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BkiUdvE4eIfiUWiFdbiw | \subsection{Isolated S(Se)-edge band in MoSe$_{2}$, WS$_{2}$ and WSe$_{2}$
from DFT calculations}
\begin{figure}[tbph]
\centering
\includegraphics[width=0.99\textwidth]{mos2_cmb.eps}\newline
\caption{Left: DFT MoS$_{2}$ zigzag ribbon band structure, without SOC;
Right: Zoom in of the Se-edge band bottom, with SOC. }
\end{figure}
\begin{figure}[tbph]
\centering
\includegraphics[width=0.99\textwidth]{mose2_cmb.eps}\newline
\caption{ Left: DFT MoSe$_{2}$ zigzag ribbon band structure, without SOC;
Right: Zoom in of the Se-edge band bottom, with SOC. }
\end{figure}
\begin{figure}[tbph]
\centering
\includegraphics[width=0.99\textwidth]{ws2_cmb.eps}\newline
\caption{ Left: DFT WS$_{2}$ zigzag ribbon band structure, without SOC;
Right: Zoom in of the S-edge band bottom, with SOC. }
\end{figure}
\end{document}
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BkiUdb44dbjiU9oEeWKt | \section{Introduction}
\smallskip
The question of existence of conformal metrics of constant or more generally prescribed curvature on riemannian manifolds is a recurrent problem in differential geometry and geometric analysis. Indeed a positive or a negative answer to such a question has far reaching consequences on the geometry and topology of the underlying manifold.
The Poincar\'e uniformization's theorem on closed surfaces, the Yamabe problem on riemannian manifolds of dimension $n \geq 3$ and the Nirenberg's problem on standard spheres $\mathbb{S}^n$, just to name a few, are well known and well studied mathematical problems. \\
A similar question, which goes back to Picard \cite{Picard1, Picard2} deals with the existence of conformal metrics of constant or prescribed curvature on surfaces with conical singularities. After the pioneering work of Picard at the beginning of the last century, such a problem has been systematically investigated by Berger \cite{Berger}, McOwen\cite{McOwen, McOwen2} and Troyanov\cite{T,T2}. More recently Bartolucci-deMarchis-Malchiodi\cite{BdM} used a Morse theoretical approach to prove further existence and multiplicity results.\\
In this paper, a first part of a series of papers, we address the problem of existence of conformal conical metrics of constant, or more generally prescribed $Q-$curvature on four dimensional riemannian manifolds. In the following we will explain in some details the geometric context of such a problem: \\
Given $(M,g)$ a compact four-dimensional Riemannian manifold, the $Q$-curvature and the Paneitz operator are defined respectively, by
\begin{equation}\label{Q-curvature}
Q_g=-\frac{1}{12}\big(\Delta_g R_g-R_g^2+3|\rm{ Ric}_g|^2\big),
\end{equation}
\begin{equation}\label{Peneitz}
P_g \varphi=\Delta_g^2\varphi+{\rm div}_g\Big(\big(\frac{2}{3}R_g g-2{\rm Ric}_g\big)\nabla u\Big),
\end{equation}
where ${\rm Ric}_g$ is the Ricci tensor and $R_g$ is the scalar curvature of $(M,g)$. \\
Similar to second order equations a natural question is the following uniformization statement: \textit{given a four-dimensional Riemannian manifold $(M,g)$, is there a metric $\tilde{g}=e^{2u}g$ in the conformal class of $g$ with constant $Q$-curvature?}
Under the conformal change of metric above, the Paneitz operator is conformally covariant:
\begin{equation}\label{conformal-P}
P_{\tilde{g}}\varphi=e^{-4u}P_g\varphi,
\end{equation}
and the $Q$ curvature of $\tilde{g}$ is given by
\begin{equation}\label{conformal-Q}
P_gu+2Q_g=2Q_{\tilde{g}}e^{4u}.
\end{equation}
From (\ref{conformal-P}) and (\ref{conformal-Q}), the question above is equivalent to the existence of solution to this fourth order equation:
\begin{equation}\label{Q-equation-reg}
P_gu+2Q_g=2\bar{Q}e^{4u},
\end{equation}
where $\bar{Q}$ is a real constant.\\
Integrating with respect to the volume element ${\rm d}V_g$, we can see that
\begin{equation*}
\kappa_P=\int_{M}Q_g{\rm d}V_g
\end{equation*}
is a constant in the conformal class of $g$, here we also point out the Gauss-Bonnet-Chern formula that links the local curvature to the global topology of $M$ is:
\begin{equation*}
\int_{M}\Big(Q_g+\frac{1}{8}|W_g|^2\Big){\rm d}V_g=4\pi^2\chi_M,
\end{equation*}
where $W_g$ denotes the Weyl's tensor of $(M,g)$ and $\chi_M$ is the Euler characteristic of $M$. From this equality and the aforementioned conformal covariance property it is not hard to imagine that $P_g$ and $Q_g$ are related to a number of studies such as Moser-Trudinger type inequatilities, log-determinant formulas and the compactification of locally flat manifolds, see \cite{Beckner,Branson-Chang-Yang,Branson-Oersted,Chang-Qing-Yang-Invent,Chang-Qing-Yang-Duke,Chang-Yang}. In many of these studies the kernel of $P_g$ is usually assumed to consist only of constants:
\begin{equation*}\label{P-assumption}
{\rm Ker}\,(P_g)=\{constants\}. \leqno(P)
\end{equation*}
In this paper consider the following prescribed $Q$-curvature equation involving singular sources
\begin{equation}\label{Q-singular}
P_gu+2Q_g=2h e^{4u}-8\pi^2\sum_{j=1}^{N}\gamma_j\Big(\delta_{q_j}-\frac{1}{{\rm vol}_g(M)}\Big),
\end{equation}
where $h$ is a smooth positive function, $N\in \mathbb N$ is a positive integer, $q_1,\cdots,q_N$ are distinct points on $M$ and where Dirac measures $\delta_{q_j}$ are located, $\gamma_j>-1$ are constants.\\
Solutions to \eqref{Q-singular} have the following geometric interpretation: Setting $\tilde{g}:= e^{2u} g$ we obtain a metric conformal to $g$ on $M \setminus \{q_1, \cdots,q_N \}$ which has a conical singularity at each $q_i$. One says that $\tilde{g}$ is represented by the divisor $D:= \sum_{i=1}^{N} \gamma_i q_i$. See Fang-Ma \cite{Fang-Ma}. Furthermore due to Gauss-Bonnet-Chern formula for conic four manifolds, see \cite{Chang-Qing-Yang-Duke}, \cite{Chang-Qing-Yang-Invent},\cite{BN19}, we have that
$$
\tilde{\kappa}_P \, := \int_{M}Q_{\tilde{g}}{\rm d}V_{\tilde{g}} \, = \, \int_{M}Q_g{\rm d}V_g \, + \, 8 \pi^2 \sum_{i=1}^{N} \gamma_i
$$
is a conformal invariant.\\
Considerable progress has been made for the regular case of (\ref{Q-singular}), that is $N=0$ in (\ref{Q-singular}). Under the assumption that the Kernel of the Paneitz operator contains only constants, Chang-Yang \cite{Chang-Yang} proved existence for $\kappa_P<8\pi^2$, Djadli-Malchiodi \cite{Djadli-Malchiodi-2008} settled the case that $\kappa_P\neq 8\pi^2n$ for any $n\in \mathbb N$. Li-Li-Liu \cite{Li-Li-Liu} gave a necessary condition for existence in the case $\kappa_P = 8 \pi^2$, Ahmedou-Ndiaye\cite{Ahmedou-Ndiaye} developed a Morse theory the case of $\kappa_P=8\pi^2n$ and Ndiaye \cite{Ndiaye-2} combined the celebrated topological argument of Bahri-Coron \cite{Bahri-Coron} with the \emph{critical point theory at infinity } in \cite{Ahmedou-Ndiaye} to derive some existence results. We point out that an essential estimate related to the proof in \cite{Djadli-Malchiodi-2008} is a priori estimate when $\kappa_P$ is away from $8\pi^2 \mathbb N$ proved by Malchiodi in \cite{Malchiodi}.
Later in \cite{Druet-Robert}, Druet and Robert extended such an a priori estimate to the following more general equation in the same class:
\begin{equation}\label{Q-equ-gen-reg}
P_g u+2b=2he^{4u},
\end{equation}
where $b$ is a smooth function. If $b=Q_g$ is the $Q$-curvature of the conformal metric $e^{2u}g$. More specially, assuming $h_k\to h_0$, $h_k\geq c_0>0$ and $b_k \to b_0$, then any sequence of solutions $\{u_k\}$ of (\ref{Q-equ-gen-reg}) with $h=h_k$ and $b=b_k$ is uniformly bounded under the condition $\int_{M}b_0{\rm d}V_g\neq8\pi^2n$, see also Malchiodi \cite{Malchiodi}.
However, bubbling can occur when $\int_{M}b_0{\rm d}V_g=8\pi^2n$ for some positive integer $n$. The understanding of this bubbling phnomenon is vital for the existence problem. The study of the blow-up profile and other blow-up phenomena for the Paneitz operator and other elliptic equations has attracted much interest recently and the reference is too numerous to be mentioned, we just list a few closely related to our article in our humble opinion: \cite{Struwe-Robert, Brendle,Chang-Qing-Yang-3,Djadli-Malchiodi-2005,Fefferman,Gursky-Viaclovsky, hyder, Li-Li-Liu,Malchiodi-2,Malchiodi-Struwe,Ndiaye,Qing-Raske,Wei-1996,Wei-Xu,zhang-weinstein}. Particularly, in \cite{zhang-weinstein}, the third named author and Weinstein have obtained sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions to (\ref{Q-equ-gen-reg}) with $h=h_k$ and $b=b_k$ and standard bubbles, and obtained the vanishing rate under the assumption that $(M,g)$ may not be locally conformally flat.
\medskip
When taking the singularities into the account as in (\ref{Q-singular}), we consider the following more general singular equation:
\begin{equation}\label{Q-equ-gen-singular}
P_gu+2b=2he^{4u}-8\pi^2\sum_{j=1}^{N}\gamma_j\Big(\delta_{q_j}-\frac{1}{vol_g(M)}\Big),
\end{equation}
where $h$ is a positive smooth function on $M$ and $b\in C^1(M)$.
Before stating our first main result, we define a {\itshape critical set} $\Gamma$ as follows:
\begin{equation*}
\Gamma=\Big\{16\pi^2n+16\pi^2\sum_{j\in J}(1+\gamma_j):\ \,n\in\mathbb{N}\cup\{0\}\ \,{\rm and}\ \,J\subset\{1,\cdots,N\}\Big\}.
\end{equation*}
In order to obtain the a priori estimates and existence results, we mainly study the blow-up phenomena for (\ref{Q-equ-gen-singular}). Let us consider the following equations:
\begin{equation}\label{Q-equation-blowup}
P_gu_k+2b_k=2h_ke^{4u_k}-8\pi^2\sum_{j=1}^{N}\gamma_j\Big(\delta_{q_j}-\frac{1}{{\rm vol}_g(M)}\Big)\quad {\rm in }\ \, M,
\end{equation}
with normalized total integration:
\begin{equation}\label{volume-normal}
\int_M e^{4u_k}{\rm d}V_g=1.
\end{equation}
Let $\{u_k\}$ be a sequence of solutions of (\ref{Q-equation-blowup}). We say $p$ is a blowup point of $u_k$ if there exists a sequence $p_k\to p$ such that
$$u_k(p_k)+8\pi^2\sum_{j=1}^N\gamma_jG(p_k,q_j)\to \infty. $$ $u_k$ is called a sequence of blowup solutions if it has a blowup point. Here $G(x,p)$ is the Green's function of $P_g$ defined in (\ref{Green-func-expression}).
For blowup solutions we assume that coefficient functions are regular enough to have limits:
\begin{equation}\label{assumption-coe}
\parallel b_k-b_0\parallel_{C^1(M)}\to 0,\quad \parallel h_k-h_0\parallel_{C^1(M)}\to 0,\quad 0<c_0<h_0<1/c_0.
\end{equation}
Without loss of generality, we assume the integration of $h_ke^{u_k}$ is $1$:
Our first main result asserts that a priori estimate holds for $u_k$, as long as $2\int_Mb_k$ does not tend to the following critical set:
\begin{equation*}
\Gamma=\Big\{16\pi^2n+16\pi^2\sum_{j\in J}(1+\gamma_j):\ \,n\in\mathbb{N}\cap\{0\}\ \,{\rm and}\ \,J\subset\{1,\cdots,N\}\Big\}.
\end{equation*}
\begin{thm}[A Priori Estimate]\label{thm-apriori-est}
Suppose (P) holds, $b$ and $h$ satisfy (\ref{assumption-coe}). If $\{u_k\}$ is a sequence of solutions of (\ref{Q-equation-blowup}) under restriction (\ref{volume-normal}) and $\int_M2b_0{\rm d}V_g\in\mathbb{R}^+\setminus \Gamma$,
$$\Big|u_k(x)+8\pi^2\sum_{j=1}^N\gamma_j G(x,q_j) \Big|\le C,\quad \forall x\in M $$
for some $C>0$ independent of $k$.
\end{thm}
In particular, the a priori estimate holds for the singular prescribing $Q$-curvature equations. Indeed Theorem \ref{thm-apriori-est} is an extension of previous results of Malchiodi \cite{Malchiodi}, Druet-Robert \cite{Druet-Robert} and Fardoun-Regbaoui \cite{fard} for the regular prescribed $Q$-curvature equation. We point out that the argument in the regular case uses in a crucial way the explicit form of the bubble, while our argument uses only the asymptotic behavior of the bubble and is based on a Pohozaev identity for equations under conformal normal coordinates (see \cite{zhang-weinstein}).\\
One indispensable part of the blowup analysis for the Q curvature equation is the classification of global solutions on $\mathbb R^4$. For this purpose we consider
the limiting equation used to describe the profile of bubbling solutions:
\begin{equation}\label{equ-liou-2}
\Delta^2 u(x)=6|x|^{4\gamma}e^{4u(x)}, \quad {\rm in} \ \; \mathbb{R}^4, \qquad
|x|^{4\gamma}e^{4u(x)}\in L^1(\mathbb{R}^4).
\end{equation}
Clearly if $u$ is a solution of (\ref{equ-liou-2}), so is $u_{\lambda}$ defined by
\begin{equation}\label{rescale}
u_{\lambda}(x)=u(\lambda x)+(1+\gamma)\log \lambda
\end{equation}
for any given $\lambda>0$. Our next main result is
\begin{thm}\label{thm-classification}
Suppose that $u$ is a solution of (\ref{equ-liou-2}) with $\gamma>-1$ and $|u(x)|=o(|x|^2)$ at infinity. Then
\begin{itemize}
\item [(i)]
$\int_{\mathbb{R}^4}6|y|^{4\gamma} e^{4u(y)}{\rm d}y=16\pi^2(1+\gamma)$.
\item [(ii)]
\begin{equation}
\begin{split}
u(x)=&\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y+C_0 \\
=&-2(1+\gamma)\log|x|+c+O(\frac{1}{|x|}), \quad |x|>1
\end{split}
\end{equation}
for some $C_0,c\in \mathbb R$,
\begin{equation}
\left\{\begin{array}{ll}
-\Delta u(x)=\frac{4(1+\gamma)}{|x|^2}+O(\frac{1}{|x|^{2+\tau}}), \\
-\frac{\partial}{\partial x_i}\Delta u(x)=-8(1+\gamma)\frac{x_i}{|x|^4}+O(\frac{1}{|x|^{3+\tau}}),\\
-\frac{\partial^2}{\partial x_i\partial x_j}\Delta u(x)=O(\frac{1}{|x|^4}),
\end{array}
\right.
\end{equation}
where $\tau=1$ if $\gamma>-3/4$, and $\tau\in(0,1)$ if $-1<\gamma\leq-3/4$.
\item [(iii)]
Furthermore, if $-3/4<\gamma<0$, $u$ is radially symmetric about the origin and is unique up to scaling in (\ref{rescale}).
\end{itemize}
\end{thm}
Theorem \ref{thm-classification} is a quantization result that also determines the asymptotic behavior of $u$ at infinity under a sub-quadratic growth condition. If
the $o(|x|^2)$ assumption is removed, we have
\begin{thm}\label{thm-classification-2}
Let $u$ be a solution of (\ref{equ-liou-2}) with $\gamma>-1$. Then after an orthogonal transformation, $u(x)$ can be represented by
\begin{equation}\label{rep-u-2}
\begin{split}
u(x)=&\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y-\sum_{j=1}^{4}a_j(x_j-x_j^0)^2+c_0 \\
=&-\sum_{j=1}^{4}a_j(x_j-x_j^0)^2-2(1+\gamma)\log|x|+c_0+O(|x|^{-\tau})
\end{split}
\end{equation}
for some $\tau>0$ and large $|x|$. The function $\Delta u$ satisfies
\begin{equation}\label{lap-u-2}
\Delta u(x)=-\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}|y|^{4\gamma}e^{4u(y)}{\rm d}y-2\sum_{j=1}^{4}a_j,
\end{equation}
where $a_j\geq 0$, $c_0$ are constants and $x^0=(x_1^0,\cdots,x_4^0)\in\mathbb{R}^4$.
Moreover, if $-3/4<\gamma<0$, $u$ is symmetric with respect to the hyperplane $\{x:x_i=0\}$ when $a_i x_i^0=0$. In particular, under the assumption $-3/4<\gamma<0$, if $a_i x_i^0=0$ for all $i=1,\cdots,4$, $u$ is radially symmetric with respect to the origin.
\end{thm}
When we were about to finish writing this article we heard that Jevnikar-Sire-Yang \cite{yang-wen} are working on a similar project independently and their results are to be posted soon.
Here we briefly outline the strategy of the proofs in our paper. For the proof of the classification result for globally defined singular equation, we follow the argument of Lin \cite{lin-classification} but we need to take care of all the complications caused by the singular source. In particular we are able to prove the complete classification for $\gamma\in (-\frac 34,0]$ and a quantization result for all $\gamma>-1$. For blowup solutions we first use a small-energy lemma (Lemma \ref{lem-small-mass-regular}) to prove that there are at most finite blowup points. Then we take advantage of a Pohozaev identity established by Weinstein-Zhang \cite{zhang-weinstein} to describe a precise asymptotic behavior of blowup solutions around a blowup point. Then the total integration as well as precise asymptotic behavior of solutions can be further determined. With this information the critical set $\Gamma$ can be identified and if the total integral of $b$ is not equal to a number corresponding to the critical set we obtain a priori estimate. Here it is worth to remark that Theorem \ref{SHT} asserts that if the strength of the singular source is not an integer, the blowup solutions satisfy a \emph{spherical Harnack inequality} around $q$, this is a new phenomenon and we shall explore in a forthcoming work. The idea of the proof of such a spherical Harnack inequaliy is as follows: If the spherical harnack inequality is violated,
there should be finite small bubbling circles around the singular
source all tending to the singular source. Around each tiny bubbling
disk there is a Pohozaev identity, and a "big" circle that includes
all these tiny disks also has a Pohozaev identity. The comparison of
these Pohozaev identities implies that the strength of the singular
source has to be an integer.
\bigskip
The organization of the remainder of this paper is as follows. In section \ref{entire} we analyze the globally defined solutions and proved the quantization and the classification results stated in Theorem \ref{thm-classification}. Then in Section \ref{preliminaries}, we list some useful facts about the conformal normal coordinates and Pohozaev identity and in section \ref{blowup-local}, we perform a blow-up analysis near singular points. Section \ref{CC-Apriori} is devoted to a concentration-compactness theorem and a priori estimate for the singular prescribing $Q$-curvature equation on 4-manifolds while in section \ref{harnack} we prove our spherical Harnack inequality. Finally we provide is the appendix an useful estimate of the difference between the geodesic distance and the Euclidean one for nearby points on the manifold.
\section[Entire solutions]{Entire solutions of fourth order singular Liouville type equations in $\mathbb{R}^4$}\label{entire}
In this section, we will follow the argument of Lin \cite{lin-classification} to analyze solutions of (\ref{equ-liou-2}) and prove Theorem \ref{thm-classification} and Theorem \ref{thm-classification-2}.
\subsection{Asymptotic behavior of entire solution}
\quad
Our argument is progressive in nature and we shall obtain a rough estimate of $u$ at infinity. For this purpose we set
\begin{equation}\label{v-def}
v(x):=\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y,
\end{equation}
which is obviously a solution of
\begin{equation}\label{v-equ}
\Delta^2v(x)=-6|x|^{4\gamma}e^{4u(x)},\quad{\rm in} \ \; \mathbb{R}^4.
\end{equation}
The asymptotic behavior of $u$ has a large to do with that of $v$, so in the first lemma we derive a rough upper bound of $v$.
For convenience we set
\begin{equation}\label{energy-R4}
\alpha=\frac{3}{4\pi^2}\int_{\mathbb R^4}|y|^{4\gamma}e^{4u}dy.
\end{equation}
\begin{lem}\label{lem-v-upper}
Suppose $u$ is a solution of (\ref{equ-liou-2}) and let $\alpha$ be given as in (\ref{energy-R4}). Then
\begin{equation}\label{v-upper}
v(x)\leq\alpha\log|x|+C
\end{equation}
for some constant $C$.
\end{lem}
\begin{proof}[\textbf{Proof}]
Since the goal is to describe asymptotic behavior it is natural to assume $|x|>4$. For such $x$, we decompose $\mathbb{R}^4=A_1\cup A_2$, where
\begin{equation*}
A_1=\Big\{y:|y-x|\leq\frac{|x|}{2}\Big\},\quad A_2=\Big\{y:|y-x|\geq\frac{|x|}{2}\Big\}.
\end{equation*}
For $y\in A_1$, $\log\frac{|x-y|}{|y|}\leq 0$ because $|y|\geq|x|-|x-y|\geq\frac{|x|}{2}\geq|x-y|$, Thus
\begin{equation*}
\int_{A_1}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma} e^{4u(y)}{\rm d}y\leq 0
\end{equation*}
and
\begin{equation*}
v(x)\leq\frac{3}{4\pi^2}\int_{A_2}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma} e^{4u(y)}{\rm d}y\leq 0.
\end{equation*}
To evaluate the integral over $A_2$, we first make two trivial observations:
$$|x-y|\leq|x|+|y|\leq|x||y|, \quad \mbox{if} \quad |y|\ge 2, $$
$$\log|x-y|\leq\log|x|+C, \quad \mbox{if }\quad |y|\leq 2 $$
where $|x|>4$ is used. Consequently
\begin{equation*}
\begin{split}
v(x)&\leq\frac{3}{4\pi^2}\int_{A_2}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma} e^{4u(y)}{\rm d}y\\
&\leq\frac{3}{4\pi^2}\Big\{\log|x|\int_{A_2\cap\{|y|\geq2\}}|y|^{4\gamma} e^{4u}{\rm d}y+\int_{A_2\cap\{|y|\leq2\}}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma} e^{4u}{\rm d}y\Big\}\\
&\leq\frac{3}{4\pi^2}\Big\{\log|x|\int_{A_2}|y|^{4\gamma} e^{4u}{\rm d}y+C\int_{A_2\cap\{|y|\leq2\}}|y|^{4\gamma} e^{4u}{\rm d}y\\
&\qquad\quad-\int_{A_2\cap\{|y|\leq2\}}\big(\log|y|\big)|y|^{4\gamma} e^{4u}{\rm d}y\Big\}\\
&\leq\frac{3}{4\pi^2}\log|x|\int_{\mathbb{R}^4}|y|^{4\gamma} e^{4u}{\rm d}y+C.
\end{split}
\end{equation*}
Lemma \ref{lem-v-upper} is established.
\end{proof}
Before proving a lower bound of $v(x)$ we derive an expression of $\Delta u(x)$ in integral form.
\begin{lem}\label{lem-lap-u}
Suppose $u$ is a solution of (\ref{equ-liou-2}). Then there exists a constant $C_1\geq 0$ such that
\begin{equation}\label{lap-u}
\Delta u(x)=-\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}|y|^{4\gamma}e^{4u(y)}{\rm d}y-C_1.
\end{equation}
\end{lem}
\begin{proof}[\textbf{Proof}]
Let $w(x)=u(x)+v(x)$. Then from the equations of $u$ and $v$ in (\ref{equ-liou-2}) and (\ref{v-equ}), we have $\Delta^2w=0$ in $\mathbb{R}^4$. Hence, $\Delta w$ is a harmonic function in $\mathbb{R}^4$. By the mean value property of harmonic functions, we have, for any $x_0\in \mathbb{R}^4$ and $r>0$,
\begin{equation*}
\begin{split}
\Delta w(x_0)=\frac{2}{\pi^2 r^4}\int_{B(x_0,r)}\Delta w(y){\rm d}y=\frac{2}{\pi^2 r^4}\int_{\partial B(x_0,r)}\Delta w(y){\rm d}\sigma,
\end{split}
\end{equation*}
where $\frac{\pi^2}{2}$ is the volume of the unit ball. That is
\begin{equation}\label{lap-w}
\frac{r}{4}\Delta w(x_0)=\dashint_{|y-x_0|=r}\frac{\partial w}{\partial r}(y){\rm d}\sigma,
\end{equation}
where $\dashint_{|y-x_0|=r}f(y){\rm d}\sigma=\frac{1}{2\pi^2r^3}\int_{|y-x_0|=r}f(y){\rm d}\sigma$ denotes the integral average of $f$ over $\partial B(x_0,r)$. Then integrating the identity above along $r$, we get
\begin{equation*}
\frac{r^2}{8}\Delta w(x_0)=\dashint_{|y-x_0|=r}w{\rm d}\sigma-w(x_0).
\end{equation*}
Therefore, the Jensen inequality implies
\begin{equation*}
\begin{split}
\exp\Big(\frac{r^2}{2}\Delta w(x_0)\Big)&\leq e^{-4w(x_0)}\exp\Big(4\dashint_{|y-x_0|=r} w {\rm d}\sigma\Big)\\
&\leq e^{-4w(x_0)}\dashint_{|y-x_0|=r} e^{4w} {\rm d}\sigma
\end{split}
\end{equation*}
From Lemma \ref{lem-v-upper}, we have $w(x)=u(x)+v(x)\leq u(x)+\alpha\log|x|+C$, and as a consequence
\begin{equation*}
\begin{split}
&\int_{0}^{\infty}r^{3-4\alpha+4\gamma}\exp\big(\frac{\Delta w(x_0)}{2}r^2\big){\rm d}r
\leq\int_{\mathbb{R}^4}|x|^{-4\alpha+4\gamma}e^{-4w(x_0)}e^{4w(x)}{\rm d}x\\
\leq& C\int_{\mathbb{R}^4}|x|^{-4\alpha+4\gamma}e^{4u(x)}|x|^{4\alpha}{\rm d}x
=C\int_{\mathbb{R}^4}|x|^{4\alpha}e^{4u}{\rm d}x<+\infty,
\end{split}
\end{equation*}
which means
$$r^{3-4\alpha+4\gamma}\exp\Big(\frac{\Delta w(x_0)}{2}r^2\Big)\in L^1\big([1,+\infty)\big).$$
From here we see $\Delta w(x_0)\leq 0$ for all $x_0\in\mathbb{R}^4$. Liouville's Theorem implies that there exists some constant $C_1\geq0$ such that $\Delta w(x)\equiv-C_1$ in $\mathbb{R}^4$.
Lemma \ref{lem-lap-u} is established based on this and
\begin{equation*}
\Delta v(x)=\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}|y|^{4\gamma}e^{4u(x)}{\rm d}y.
\end{equation*}
\end{proof}
With the help of the representation for $\Delta u$, we can estimate $v(x)$ from below for $|x|$ large. We will use following result in Lemma 2.3 of \cite{lin-classification}.
\medskip
Let $h(x)$ be the solution of
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta^2 h(x)=f(x), && {\rm in} \ \; \Omega,
\\
\Delta h(x)=h(x)=0, && {\rm on} \ \; \partial\Omega,
\end{array}
\right.
\end{equation*}
where $\Omega$ is a bounded domain of $\mathbb{R}^4$.
\begin{lemA}\label{lem-BM}\cite{lin-classification}
Suppose $f\in L^1(\bar{\Omega})$.Then for any $\delta\in(0,32\pi^2)$, there exists a constant $C_{\delta}>0$ such that
\begin{equation*}
\int_{\Omega}\exp\Big(\frac{\delta|h|}{\parallel f\parallel_{L^1}}\Big){\rm d}x\leq C_{\delta}(diam \; \Omega)^4,
\end{equation*}
where $diam \;\Omega$ denotes the diameter of $\Omega$.
\end{lemA}
\begin{lem}\label{lem-v-lower}
Let $u$ be a solution of (\ref{equ-liou-2}) and $v$ be in (\ref{v-def}). Then for given $\varepsilon>0$, there exists a constant $R=R(\varepsilon)$ only depending on $\varepsilon$ such that
\begin{equation}\label{v-lower}
v(x)\geq(\alpha-\varepsilon)\log|x|, \quad |x|>R(\epsilon).
\end{equation}
\end{lem}
\begin{proof}[\textbf{Proof}]
We first prove a claim slightly weaker than (\ref{v-lower}): for any $\varepsilon>0$, there exists $R=R(\varepsilon)>0$ such that
\begin{equation}\label{v-lower-rough}
v(x)\geq(\alpha-\frac{\varepsilon}{2})\log|x| +\frac{3}{4\pi^2}\int_{B(x,1)}(\log|x-y|)|y|^{4\gamma}e^{4u(y)}{\rm d}y.
\end{equation}
To prove (\ref{v-lower-rough}) we consider $\mathbb R^4$ as a disjoint union of three sets: $\mathbb{R}^4=A_1\cup A_2\cup A_3$, where
\begin{align*}
A_1&=\{y:|y|<R_0\}, \\
A_2&=\{y:|x-y|\leq |x|/2,|y|\geq R_0\}, \\ A_3&=\{y:|x-y|\geq |x|/2,|y|\geq R_0\}.
\end{align*}
Then we choose $R_0=R_0(\varepsilon)$ sufficiently large so that
\begin{equation*}
\begin{split}
&\frac{3}{4\pi^2}\int_{A_1}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y-\alpha \log|x|\\
=&\frac{3}{4\pi^2}\log|x|\int_{A_1}\frac{\log|x-y|-\log|x|-\log|y|}{\log|x|}|y|^{4\gamma}e^{4u(y)}{\rm d}y-\frac{\varepsilon}{8} \log|x| \\
\geq& -\frac{\varepsilon}{4} \log|x|
\end{split}
\end{equation*}
for large $|x|$. Thus we have
\begin{equation}\label{int-A1}
\frac{3}{4\pi^2}\int_{A_1}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y\geq (\alpha-\frac{\varepsilon}{4} )\log|x|.
\end{equation}
For $x\in A_2$ and $|x|$ large, we have $\frac{|x|}{2}\leq |x|\leq\frac{3}{2}|x|$. Then
\begin{equation}\label{int-A2}
\begin{split}
&\int_{A_2}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y \\
=&\int_{A_2}(\log|x-y|)|y|^{4\gamma}e^{4u(y)}{\rm d}y-\int_{A_2}(\log|y|)|y|^{4\gamma}e^{4u(y)}{\rm d}y \\
\geq& \int_{B(x,1)}(\log|x-y|)|y|^{4\gamma}e^{4u(y)}{\rm d}y-\log(2|x|)\int_{A_2}|y|^{4\gamma}e^{4u(y)}{\rm d}y.
\end{split}
\end{equation}
For $y\in A_3$, we use two trivial inequalities: $|x-y|\geq\frac{|x|}{2}\geq\frac{|y|}{4}$ if $|y|\le 2|x|$ and
$|x-y|\geq|y|-|x|\geq\frac{|y|}{2}$ if $|y|\ge 2|x|$. Clearly in both cases, we have
\begin{equation*}
\frac{|x-y|}{|y|}\geq\frac{1}{4},\quad y\in A_3.
\end{equation*}
Therefore,
\begin{equation}\label{int-A3}
\frac{3}{4\pi^2}\int_{A_3}\log\big(\frac{|x-y|}{|y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y\geq \log\frac{1}{4}\int_{A_3}|y|^{4\gamma}e^{4u(y)}{\rm d}y.
\end{equation}
From (\ref{int-A1}), (\ref{int-A2}), (\ref{int-A3}) and $|y|^{4\gamma}e^{4u(y)}\in L^1(\mathbb{R}^4)$, we obtain (\ref{v-lower-rough}).
\medskip
Next, we show that
\begin{equation}\label{int-B_x1-v}
\int_{B(x,1)}(\log|x-y|)|y|^{4\gamma}e^{4u(y)}{\rm d}y\geq-C
\end{equation}
for some positive constant $C$. For this purpose we set
\begin{equation}
\tilde{u}(x)=u(x)-\gamma\log |x|
\end{equation}
Then $\tilde{u}$ satisfies
\begin{equation}\label{equ-liou}
\left\{\begin{array}{lcl}
\Delta^2 \tilde{u}(x)=6e^{4\tilde{u}(x)}-8\pi^2\gamma\delta_0, && {\rm in} \ \; \mathbb{R}^4,
\\
e^{4\tilde{u}}\in L^1(\mathbb{R}^4). && \quad
\end{array}
\right.
\end{equation}
Let $0<\varepsilon_0<\pi^2$ and $R_0=R_0(\varepsilon_0)$ be sufficiently large such that
\begin{equation}\label{energy-B4}
6\int_{B(x,4)}|y|^{4\gamma}e^{4u(y)}{\rm d}y=6\int_{B(x,4)}e^{4\tilde{u}(y)}{\rm d}y\leq\varepsilon_0, \quad {\rm for}\ \, |x|\geq R_0,
\end{equation}
then we let $h$ be the solution of
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta^2 h(x)=6e^{4\tilde{u}(y)}, && {\rm in} \ \; B(x,4),
\\
h(x)=\Delta h(x)=0, && {\rm on} \ \; \partial B(x,4).
\end{array}
\right.
\end{equation*}
From Lemma \ref{lem-BM}, we can see that for $\varepsilon_0>0$ small,
\begin{equation}\label{int-h-B4}
\int_{B(x,4)}e^{24|h|}{\rm d}y\leq c_1
\end{equation}
for some constant $c_1$ independent of $x$.
Next we set $q(y)=\tilde{u}(y)-h(y)$ for $y\in B(x,4)$, which clearly satisfies
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta^2 q(y)=0, && {\rm in} \ \; B(x,4),
\\
q(y)=\tilde{u}(y),\quad \Delta q(y)=\Delta\tilde{u}(y), && {\rm on} \ \; \partial B(x,4).
\end{array}
\right.
\end{equation*}
Let $\tilde{q}(y)=-\Delta q(y)$. Then by Lemma \ref{lem-lap-u}, we see that for $|x|$ large enough and $y\in\partial B(x,4)$
\begin{equation*}
\begin{split}
\tilde{q}(y)=&-\Delta \tilde{u}(y)=-\Delta u(y)+\gamma\Delta(\log |y|)=-\Delta u(y)-\frac{2\gamma}{|y|^2}.
\end{split}
\end{equation*}
By setting
\begin{equation*}
\hat{q}(y)=\tilde{q}(y)+\frac{2\gamma}{|y|^2}=-\Delta u(y)=\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|z-y|^2}|z|^{4\gamma}e^{4u(z)}{\rm d}z+C_1,
\end{equation*}
we obviously have $\hat{q}(y)>0$ on $\partial B(x,4)$, and hence $\tilde{q}(y)>-2\gamma/|y|^2$ on $\partial B(x,4)$. Observing that $1/|y|^2$ is the fundamental solution of $\Delta$. In other words, $\hat{q}(y)$ is harmonic in $B(x,4)$ with positive boundary value on $\partial B(x,4)$. The maximum principle implies $\hat{q}>0$ in $B(x,4)$. Thus, by the Harnack inequality and mean value property of harmonic functions, we have
\begin{equation}\label{est-lap-q}
\begin{split}
\tilde{q}(y)&=\hat{q}(y)-\frac{2\gamma}{|y|^2} \\
&\leq c_2\hat{q}(x)-\frac{2\gamma}{|y|^2}
=-\dashint_{\partial B(x,4)}\Delta u{\rm d }\sigma-\frac{2\gamma}{|y|^2}=-\dashint_{\partial B(x,4)}\Delta \tilde{u}{\rm d }\sigma,\quad y\in\overline{B(x,2)},
\end{split}
\end{equation}
with a constant $c_2$.
\medskip
Integrating (\ref{equ-liou}) along $r$, we have
\begin{equation*}
\begin{split}
&\dashint_{\partial B(x,4)}\Delta \tilde{u}{\rm d}\sigma-\Delta\tilde{u}(x)=\int_{0}^{r}\frac{3}{\pi^2 s^3}\int_{0}^{s}\int_{\partial B(x,t)}e^{4\tilde{u}}{\rm d}\sigma{\rm d}t{\rm d}s \\
=&\int_{0}^{r}\frac{3}{\pi^2 s^3}\int_{0}^{s}t^3\int_{\partial B(x,1)}e^{4\tilde{u}}{\rm d}\sigma{\rm d}t{\rm d}s \\
=&\int_{0}^{r}t^3\int_{\partial B(x,1)}e^{4\tilde{u}}\Big(\frac{3}{2\pi^2t^2}-\frac{3}{2\pi^2r^2}\Big){\rm d}\sigma{\rm d}t.
\end{split}
\end{equation*}
That is
\begin{equation}\label{ave-int-tildeu}
\dashint_{\partial B(x,4)}\Delta \tilde{u}{\rm d}\sigma-\Delta\tilde{u}(x)=\frac{3}{2\pi^2}\int_{ B(x,r)}\Big(\frac{1}{|x-y|^2}-\frac{1}{r^2}\Big)e^{4\tilde{u}}{\rm d}y.
\end{equation}
Next by Lemma \ref{lem-lap-u} and (\ref{ave-int-tildeu}), we can see
\begin{equation*}
\begin{split}
&-\dashint_{\partial B(x,4)}\Delta \tilde{u}{\rm d}\sigma=-\Delta\tilde{u}(x)-\frac{3}{2\pi^2}\int_{ B(x,r)}\frac{1}{|x-y|^2}e^{4\tilde{u}}{\rm d}y+ \frac{3}{2\pi^2r^2}\int_{ B(x,r)}e^{4\tilde{u}}{\rm d}y\\
=&-\Delta u(x)-\frac{3}{2\pi^2}\int_{ B(x,r)}\frac{1}{|x-y|^2}e^{4\tilde{u}}{\rm d}y+ \frac{3}{2\pi^2r^2}\int_{ B(x,r)}e^{4\tilde{u}}{\rm d}y-\frac{2\gamma}{|x|^2}\\
=&\frac{3}{2\pi^2r^2}\int_{|x-y|\geq r}\frac{1}{|x-y|^2}e^{4\tilde{u}}{\rm d}y+ \frac{3}{2\pi^2r^2}\int_{ B(x,r)}e^{4\tilde{u}}{\rm d}y-\frac{2\gamma}{|x|^2}+C_1.
\end{split}
\end{equation*}
In particular, for $r=4$ and $|x|$ large,
\begin{equation}\label{est-lap-aveu}
-\dashint_{\partial B(x,4)}\Delta \tilde{u}{\rm d}\sigma\leq c_3.
\end{equation}
Hence, from (\ref{est-lap-q}), we get
\begin{equation}\label{est-lap-q-2}
\tilde{q}(y)\leq c_4,\,\quad y\in\overline{B(x,2)},
\end{equation}
and immediately
$$|\tilde{q}(y)|\leq c_5,\,\quad y\in\overline{B(x,2)}.$$
Since $q$ satisfies
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta q(y)=-\tilde{q}(y), && {\rm in} \ \; B(x,4),
\\
q(y)=\tilde{u}(y), && {\rm on} \ \; \partial B(x,4),
\end{array}
\right.
\end{equation*}
by estimates for linear elliptic equations, we have for any $p>1$ and $\sigma>2$,
\begin{equation}\label{est-sup-q}
\sup_{B(x,1)} q\leq c\big(\parallel q^{+}\parallel_{L^p(B(x,2))}+\parallel \tilde{q}\parallel_{L^{\sigma}(B(x,2))}\big),
\end{equation}
where $c=c(p,\sigma)$.
On the other hand, we observe that $q^{+}(y)\leq\tilde{u}^{+}(y)+|h(y)|$ for $y\in B(x,4)$. Then by (\ref{int-h-B4}), we get
\begin{equation*}
\begin{split}
\int_{ B(x,2)}(q^{+})^p\leq c_6\int_{ B(x,2)}e^{2q^{+}}\leq c_5\Big(\int_{ B(x,2)}e^{4\tilde{u}^{+}}\Big)^{\frac{1}{2}}\Big(\int_{ B(x,2)}e^{4|h|}\Big)^{\frac{1}{2}}\leq c_7\Big(\int_{ B(x,2)}e^{4\tilde{u}^{+}}\Big)^{\frac{1}{2}}
\end{split}
\end{equation*}
Since $e^{4\tilde{u}^{+}}\leq 1+e^{4\tilde{u}}$, we have $\parallel q^{+}\parallel_{L^p(B(x,2))}\leq c_7$, which together with (\ref{est-lap-q-2}) and (\ref{est-sup-q}) implies
\begin{equation}\label{est-sup-q-2}
\sup_{B(x,1)} q\leq c_8.
\end{equation}
In view of $\tilde{u}=h+q$, we now obtain
\begin{equation}
\tilde{u}(y)\leq h(y)+q(y)\leq c_8+|h(y)|,\,\quad y\in\overline{B(x,2)}.
\end{equation}
Therefore,
\begin{equation}\label{int-24u}
\int_{ B(x,1)}e^{24\tilde{u}}\leq c_9 \int_{ B(x,1)}e^{24|h|}{\rm d}y\leq c_{10},
\end{equation}
Then
\begin{equation*}
\Big|\int_{ B(x,1)}(\log|x-y|)e^{4\tilde{u}}{\rm d}y\Big|\leq\Big(\int_{ B(x,1)}(\log|x-y|)^2{\rm d}y\Big)^{\frac{1}{2}}\Big(\int_{ B(x,1)}e^{8\tilde{u}(y)}{\rm d}y\Big)^{\frac{1}{2}}\leq c_{11},
\end{equation*}
which means
\begin{equation}\label{int-minor-term}
\Big|\int_{B(x,1)}(\log|x-y|)|y|^{4\gamma}e^{4u(y)}{\rm d}y\Big|\leq c_{11},
\end{equation}
where $c_{11}$ is a constant independent of $x$ ($|x|$ large). As a consequence, (\ref{v-lower-rough}) and (\ref{int-minor-term}) lead to
\begin{equation*}
v(x)\geq(\alpha-\frac{\varepsilon}{2})\log|x|-c_{11}\geq(\alpha-\varepsilon)\log|x|
\end{equation*}
for $|x|$ large, which is (\ref{v-lower}).
\medskip
To estimate $\Delta v$, we observe that
\begin{equation*}
\Delta v(x)=-\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}|y|^{4\gamma}e^{4u(y)}{\rm d}y=-\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}e^{4\tilde{u}(y)}{\rm d}y.
\end{equation*}
We decompose $\mathbb{R}^4=B_1\cup B_2\cup B_3$ for $|x|$ large, where
\begin{align*}
B_1&=\{y:|x-y|\geq|x|/2\}, \\
B_2&=\{y:1\leq |x-y|\leq|x|/2,|y|\geq R_0\}, \\
B_3&=\{y:|x-y|\leq 1\}.
\end{align*}
Then
\begin{equation*}
\int_{B_1}\frac{1}{|x-y|^2}e^{4\tilde{u}(y)}{\rm d}y\leq\frac{C}{|x|^2}\to 0,\quad {\rm as}\ \, |x|\to\infty,
\end{equation*}
\begin{equation*}
\int_{B_2}\frac{1}{|x-y|^2}e^{4\tilde{u}(y)}{\rm d}y\leq\int_{B_2}e^{4\tilde{u}(y)}{\rm d}y\to 0,\quad {\rm as}\ \, |x|\to\infty,
\end{equation*}
and
\begin{equation*}
\begin{split}
&\int_{B_3}\frac{1}{|x-y|^2}e^{4\tilde{u}(y)}{\rm d}y \\
\leq&\Big( \int_{ |x-y|\leq 1}\big(\frac{1}{|x-y|}\big)^{\frac{8}{5}}{\rm d}y\Big)^{\frac{5}{8}}\Big( \int_{ |x-y|\leq 1}e^{4\tilde{u}}{\rm d}y\Big)^{\frac{1}{4}}( \int_{ |x-y|\leq 1}e^{24\tilde{u}}{\rm d}y\Big)^{\frac{1}{8}} \\
\leq& c\Big( \int_{ |x-y|\leq 1}e^{4\tilde{u}}{\rm d}y\Big)^{\frac{1}{4}}\to 0,\quad {\rm as}\ \, |x|\to\infty,
\end{split}
\end{equation*}
where we have used (\ref{int-24u}). Lemma \ref{lem-v-lower} is established.
\end{proof}
With the estimates of $v(x)$ near infinity and the expression of $\Delta u$, we can show the expression of $u$ in integral form under the condition $|u(x)|=o(|x|^2)$ at $\infty$:
\begin{lem}\label{lem-rep-u}
Suppose $|u(x)|=o(|x|^2)$ at $\infty$. Then there exists some constant $C_0$ such that
\begin{equation}\label{rep-u}
u(x)=\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y+C_0.
\end{equation}
Furthermore, for any given $\varepsilon>0$,
\begin{equation}\label{u-est}
-\alpha\log|x|-C\leq u(x)\leq(-\alpha+\varepsilon)\log|x|,\quad |x|\geq R(\varepsilon),
\end{equation}
where $R(\varepsilon)$ comes from Lemma \ref{lem-v-lower}.
\end{lem}
\begin{proof}[\textbf{Proof}]
We start from the integral expression of $\Delta u$ in Lemma \ref{lem-lap-u}:
\begin{equation*}
\Delta u(x)=-\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}|y|^{4\gamma}e^{4u(y)}{\rm d}y-C_1,\quad C_1\ge 0,
\end{equation*}
and we first prove $C_1=0$ by contradiction. If $C_1>0$ we have
\begin{equation*}
\Delta u(x)\leq -C_1<0,\quad |x|\geq R_0,
\end{equation*}
where $R_0$ is large.
Let
\begin{equation}
h(y)=u(y)+\varepsilon|y|^2+A\big(|y|^{-2}-R_0^{-2}\big).
\end{equation}
Under the assumption of $|u(y)|=o(|y|^2)$ at $\infty$, we have $\lim\limits_{|y|\to +\infty}h(y)=+\infty$ for any fixed $\varepsilon>0$ and $A>0$. So we choose $\varepsilon>0$ small to make
\begin{equation}
\Delta h(y)=\Delta u(y)+8\varepsilon<-\frac{C_1}{2}<0,\quad |y|\geq R_0,
\end{equation}
and $A$ sufficiently large such that $\inf\limits_{|y|\geq R_0} h(y)$ is achieved by some $y_0\in \mathbb{R}^4$ and $|y_0|>R_0$. Clearly we have obtained a contradiction to the maximum principle. Hence, $C_1=0$ and $u+v$ is harmonic in $\mathbb{R}^4$.
\medskip
From Lemma \ref{lem-v-upper} and Lemma \ref{lem-v-lower}, we know for $|x|$ large enough
\begin{equation*}
(\alpha-\varepsilon)\log|x|\leq v(x)\leq \alpha\log|x|+C,
\end{equation*}
which together with the assumption $|u(x)|=o(|x|^2)$ at $\infty$ indicates
\begin{equation}
|u(x)+v(x)|=o(|x|^2)\quad {\rm at} \ \,\infty.
\end{equation}
Since $u+v$ is harmonic, by the gradient estimates for harmonic functions, we have
\begin{equation*}
u(x)+v(x)=\sum_{j=1}^{4}a_j x_j+a_0
\end{equation*}
with some constants $a_j\in\mathbb{R},\,j=0,\cdots,4$. Therefore, for $|x|$ large enough, we get
\begin{equation*}
e^{4u(x)}=e^{a_0}e^{-4v(x)}e^{\sum_{j=1}^4a_j x_j}\geq C|x|^{-4\alpha}e^{\sum_{j=1}^4 a_j x_j}.
\end{equation*}
Since $|y|^{4\gamma}e^{4u(x)}\in L^1(\mathbb{R}^4)$, we have $a_j=0$ for $1\leq j\leq 4$. Therefore,
\begin{equation*}
u(x)=-v(x)+a_0=\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y+C_0,
\end{equation*}
and then
\begin{equation*}
-\alpha\log|x|-C\leq u(x)\leq(-\alpha+\varepsilon)\log|x|,
\end{equation*}
for $|x|$ large. Lemma \ref{lem-rep-u} is established.
\end{proof}
Next we need a Pohozaev identity for $u$ of
\begin{equation}\label{equ-PI}
\Delta u=Q(x)e^{4u}\ \,{\rm in} \ \,\mathbb{R}^4.
\end{equation}
\begin{lem}\label{lem-PI}
Suppose $u$ is an entire smooth solution of (\ref{equ-PI}). Then for any bounded domain, we have
\begin{equation}\label{PI-omega}
\begin{split}
&\int_{\Omega}\big(Q+\frac{1}{4}<x,\nabla Q>e^{4u}\big){\rm d}x \\
=&\frac{1}{4}\int_{\partial\Omega}<x,\nu>Q(x) e^{4u}{\rm d}\sigma+\int_{\partial\Omega}\Big\{\frac{1}{2}|\Delta u|^2<x,\nu>-2\frac{\partial u}{\partial\nu}\Delta u \\
&\quad -<x,\nabla u>\frac{\partial\Delta u}{\partial\nu}-<x,\nabla\Delta u>\frac{\partial u}{\partial \nu}+<x,\nu><\nabla u,\nabla \Delta u>\Big\}{\rm d}\sigma.
\end{split}
\end{equation}
In particular, taking $\Omega=B_R$, we have
\begin{equation}\label{PI-BR}
\begin{split}
&\int_{B_R}Q(x)e^{4u}{\rm d}x+\frac{1}{4}\int_{B_R}<x,\nabla Q>e^{4u}{\rm d}x \\
=&\frac{1}{4}\int_{\partial B_R}|x|Q e^{4u}{\rm d}\sigma+\frac{1}{2}\int_{\partial B_R}|x||\Delta u|^2{\rm d}\sigma-2\int_{\partial B_R}\frac{\partial u}{\partial r}\Delta u{\rm d}\sigma -\int_{\partial B_R}|x|\frac{\partial u}{\partial r}\frac{\partial \Delta u}{\partial r}{\rm d}\sigma .
\end{split}
\end{equation}
\end{lem}
\begin{proof}[\textbf{Proof}]
Multiplying (\ref{equ-PI}) by $x\cdot\nabla u$, we have
\begin{equation}\label{IBP}
\int_{\Omega}\nabla^2(x\cdot\nabla u)=\int_{\Omega}Q(x)e^{4u}(x\cdot\nabla u).
\end{equation}
After integrating by parts and direct computation, we get
\begin{equation*}
\begin{split}
({\rm RHS}) \; {\rm of}\; (\ref{IBP})=\frac{1}{4}\int_{\partial\Omega}<x,\nu>Q(x) e^{4u}{\rm d}\sigma-\int_{\Omega}\big(Q+\frac{1}{4}<x,\nabla Q>e^{4u}\big){\rm d}x,
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
({\rm LHS}) \; {\rm of}\; (\ref{IBP})=&-\frac{1}{2}\int_{\partial\Omega}|\Delta u|^2<x,\nu>+2\int_{\partial\Omega}\frac{\partial u}{\partial\nu}\Delta u+\int_{\partial\Omega}<x,\nabla u>\frac{\partial\Delta u}{\partial\nu}\\
&+\int_{\partial\Omega}<x,\nabla\Delta u>\frac{\partial u}{\partial \nu}-\int_{\partial\Omega}<x,\nu><\nabla u,\nabla \Delta u>.
\end{split}
\end{equation*}
Thus we establish (\ref{PI-omega}). Taking $\Omega=B_R$, we immediately obtain (\ref{PI-BR}) from (\ref{PI-omega}).
\end{proof}
From the Pohozaev identity we shall determine the exact value of $\alpha$.
\begin{lem}\label{alpha}
Let $u$ be a solution of (\ref{equ-liou-2}). Assume $|u(x)|=o(|x|^2)$ at $\infty$, then $\alpha=2(1+\gamma)$.
\end{lem}
\begin{proof}[\textbf{Proof}]
Taking $Q(x)=6|x|^{4\gamma}$ in (\ref{PI-BR}), we have
\begin{equation}\label{PI-BR-2}
\begin{split}
&6(1+\gamma)\int_{B_R}|x|^{4\gamma}e^{4u}{\rm d}x \\
=&\frac{1}{4}\int_{\partial B_R}6r^{4\gamma+1}e^{4u}{\rm d}\sigma+\frac{1}{2}\int_{\partial B_R}r|\Delta u|^2-2\int_{\partial B_R}\frac{\partial u}{\partial r}\Delta u -\int_{\partial B_R}r\frac{\partial u}{\partial r}\frac{\partial \Delta u}{\partial r} .
\end{split}
\end{equation}
In view of Lemma \ref{lem-rep-u}, we have obtained
\begin{equation*}
u(x)=\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y+C_0,
\end{equation*}
and $e^{u(y)}\geq |y|^{-4\alpha}$ for $|y|$ large enough. Then $|y|^{4\gamma}e^{4u(y)}\geq|y|^{4(\gamma-\alpha)}$. Hence $|y|^{4\gamma}e^{4u(y)}\in L^1(\mathbb{R}^4)$ implies $\alpha>1+\gamma$. On the other hand, by the representation of $u$ and direct calculations, there hold
\begin{equation*}
\frac{\partial u}{\partial r}(x)=-\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\frac{x\cdot(x-y)}{|x||x-y|^2}|y|^{4\gamma}e^{4u(y)}{\rm d}y,
\end{equation*}
\begin{equation*}
\Delta u(x)=-\frac{3}{2\pi^2}\int_{\mathbb{R}^4}\frac{1}{|x-y|^2}|y|^{4\gamma}e^{4u(y)}{\rm d}y,
\end{equation*}
and
\begin{equation*}
\frac{\partial}{\partial r}\Delta u(x)=\frac{3}{\pi^2}\int_{\mathbb{R}^4}\frac{x\cdot(x-y)}{|x||x-y|^4}|y|^{4\gamma}e^{4u(y)}{\rm d}y,
\end{equation*}
Recall the definication of $u$, Lemma \ref{lem-rep-u} and $\alpha>1+\gamma$, then we have
\begin{equation}\label{u-limit}
\begin{split}
&\lim_{r\to +\infty}\frac{\partial u}{\partial r}=0, \\
&\lim_{r\to +\infty}r\frac{\partial u}{\partial r}=-\alpha,\\
&\lim_{r\to +\infty}r^2\Delta u=-2\alpha, \\
&\lim_{r\to +\infty}r^3\frac{\partial }{\partial r}\Delta u=4\alpha,
\end{split}
\end{equation}
where $r=|x|$. Therefore, applying the Pohozaev identity (\ref{PI-BR-2}) and (\ref{u-limit}), we have
\begin{equation*}
8\pi^2(1+\gamma)\alpha=4\pi^2\alpha^2,
\end{equation*}
which leads to $\alpha=2(1+\gamma)$.
\end{proof}
Now we can determine the asymptotic behavior of $u$ at infinity using the exact value of $\alpha$.
\begin{lem}\label{lem-u-asy}
Let $u$ be a solution of (\ref{equ-liou-2}) and suppose $|u(x)|=o(|x|^2)$ at $\infty$. Then there exist constants $c$ and $a_j$ $(j=1,\cdots,4)$ such that for $|x|$ large enough,
\begin{equation}\label{u-asy}
u(x)=-2(1+\gamma)\log|x|+c+O(\frac{1}{|x|}),
\end{equation}
the derivatives of $u$ at infinity are determined in two cases: if $\gamma>-3/4$,
\begin{equation}\label{u-high-asy}
\left\{\begin{array}{ll}
-\Delta u(x)=\frac{4(1+\gamma)}{|x|^2}+\sum_{j=1}^{4}\frac{a_jx_j}{|x|^4}+O(\frac{1}{|x|^4}), \\
-\frac{\partial}{\partial x_i}\Delta u(x)=-8(1+\gamma)\frac{x_i}{|x|^4}+O(\frac{1}{|x|^4}),\\
-\frac{\partial^2}{\partial x_i\partial x_j}\Delta u(x)=O(\frac{1}{|x|^4}).
\end{array}
\right.
\end{equation}
if $-1<\gamma\leq -3/4$, there exists a constant $\tau\in(0,1)$ such that
\begin{equation}
\left\{\begin{array}{ll}
-\Delta u(x)=\frac{4(1+\gamma)}{|x|^2}+O(\frac{1}{|x|^{2+\tau}}), \\
-\frac{\partial}{\partial x_i}\Delta u(x)=-8(1+\gamma)\frac{x_i}{|x|^4}+O(\frac{1}{|x|^{3+\tau}}),\\
-\frac{\partial^2}{\partial x_i\partial x_j}\Delta u(x)=O(\frac{1}{|x|^4}).
\end{array}
\right.
\end{equation}
\end{lem}
\begin{proof}[\textbf{Proof}]
Let $w(x)=u(\frac{x}{|x|^2})-2(1+\gamma)\log|x|$, then from the equation of $u$ and the assumption, we see $w$ satisfies
\begin{equation}\label{equ-w}
\left\{\begin{array}{ll}
\Delta^2 w(x)=6|x|^{4\gamma}e^{4w(x)},\quad {\rm in} \ \, \mathbb{R}^4\setminus\{0\}, \\
|w(x)|=o(\log\frac{1}{|x|}),\quad |\Delta w(x)|=o(\frac{1}{|x|^2}),\quad{\rm as}\ \, |x|\to 0.
\end{array}
\right.
\end{equation}
Let $h(x)$ be a solution of
\begin{equation}
\left\{\begin{array}{lcl}
\Delta^2 h(x)=6|x|^{4\gamma}e^{4w(x)},&& {\rm in} \ \, B_1\setminus \{0\}, \\
h(x)=w(x),\quad \Delta h(x)=\Delta w(x),&&{\rm on} \ \, \partial B_1.
\end{array}
\right.
\end{equation}
and $q(x)=w(x)-h(x)$. Then $q(x)$ satisfies
\begin{equation}
\left\{\begin{array}{lcl}
\Delta^2 q(x)=0,&& {\rm in} \ \, B_1, \\
q(x)=\Delta q=0,&& {\rm on} \ \, \partial B_1,\\
|q(x)|=o(\log\frac{1}{|x|}),\quad |\Delta q(x)|=o(\frac{1}{|x|^2}),&& {
\rm as}\ \, |x|\to 0.
\end{array}
\right.
\end{equation}
First for $\Delta q$, since its growth near the singular source is weaker than fundamental solutions, the singularity is removable, thus $\Delta q=0$ in $B_1$.
By exactly the same reason we further conclude that $q\equiv 0$ in $B_1$. That means $w(x)=h(x)\in C^{0,\tau}(\bar{B}_1)$. It suffices to consider the regularity of $h$ in $B_1$.
Note that
\begin{equation*}
\begin{split}
|x|^{4\gamma}e^{4w(x)}&=|x|^{4\gamma}e^{4u(\frac{x}{|x|^2})-8(1+\gamma)\log|x|} \\
&\sim|x|^{4\gamma}|x|^{-8(1+\gamma)}|x|^{4\alpha} \sim|x|^{4\gamma},\quad {\rm near} \ \, 0,
\end{split}
\end{equation*}
where we used Lemma \ref{lem-rep-u} and Lemma \ref{alpha}.
\textbf{Case 1:}
$\gamma>-\frac{3}{4}$. In this case, there exists some $p>\frac{4}{3}$ such that $|x|^{4\gamma}e^{4w(x)}\in L^p(\bar{B}_1)$. By the standard elliptic estimates and general Sobolev inequality, we have $h\in C^{1,\tau}(B_1)$ for some $0<\tau<1$, which is valid for $w$ as well. Then it is not difficult to obtain (\ref{u-asy}), and (\ref{u-high-asy}) follows immediately.
\textbf{Case 2:}
$-1<\gamma\leq-\frac{3}{4}$, we can still find some $p>1$ (indeed $1<p\leq\frac{4}{3}$) such that $|x|^{4\gamma}e^{4w(x)}\in L^p(\bar{B}_1)$. Then by the regularity theorems of linear elliptic equations, $h(x)\in C^{0,\tau}(\bar{B}_1)$ for some $0<\tau<1$. In this case, we get
\begin{equation}
u(x)=-2(1+\gamma)\log|x|+c+O(\frac{1}{|x|^{\tau}}),
\end{equation}
and
\begin{equation}
\left\{\begin{array}{ll}
-\Delta u(x)=\frac{4(1+\gamma)}{|x|^2}+O(\frac{1}{|x|^{2+\tau}}), \\
-\frac{\partial}{\partial x_i}\Delta u(x)=-8(1+\gamma)\frac{x_i}{|x|^4}+O(\frac{1}{|x|^{3+\tau}}),\\
-\frac{\partial^2}{\partial x_i\partial x_j}\Delta u(x)=O(\frac{1}{|x|^4}).
\end{array}
\right.
\end{equation}
\end{proof}
\medskip
\subsection{Classification of entire solutions for the case $-3/4<\gamma<0$}
\quad
In this subsection, we will show the solution of (\ref{equ-liou}) or (\ref{equ-liou-2}) has radial symmetry and uniqueness property up to scaling if $-3/4<\gamma<0$. Similar to \cite{lin-classification}, we will use the method of moving planes.
Suppose that $u$ is a smooth entire solution of (\ref{equ-liou-2}) with $|u(x)|=o(|x|^2)$ at $\infty$. Recall $-\Delta u>0$ in $\mathbb{R}^4$ and (\ref{u-asy})$\sim$(\ref{u-high-asy}), so we will apply the method of moving planes to $-\Delta u$. Let $v(x)=-\Delta u(x)$. Then by Lemma \ref{lem-u-asy}, $v(x)$ has the following harmonic asymptotic expansion at $\infty$:
\begin{equation}\label{harmonic-asy}
\left\{\begin{array}{ll}
v(x)=\frac{4(1+\gamma)}{|x|^2}+O(\frac{1}{|x|^3}), \\
v_{x_i}(x)=-8(1+\gamma)\frac{x_i}{|x|^4}+O(\frac{1}{|x|^4}),\\
v_{x_ix_j}(x)=O(\frac{1}{|x|^4}).
\end{array}
\right.
\end{equation}
First, we state some conventional notations for moving planes. For any $\lambda\in\mathbb{R}$, let $T_{\lambda}=\{x\in\mathbb{R}^4:x_1=\lambda\}$, $\Sigma_{\lambda}=\{x:x_1>\lambda\}$ and $x^{\lambda}=(2\lambda-x_1,x_2,x_3,x_4)$ be the reflection point of $x$ with respect to $T_{\lambda}$. In addition, when applying the method of moving plane, we need two lemmas in \cite{CGS}, also can be seen in \cite{lin-classification}.
\begin{lemA}\label{CGS-lem-1}\cite{CGS}
Let $v$ be a positive function defined in a neighborhood at $\infty$ satisfying the asymptotic expansion (\ref{harmonic-asy}). Then there exists $\bar{\lambda}_0<0$ and $R>0$ such that there holds
\begin{equation*}
v(x)>v(x^{\lambda})
\end{equation*}
for $\lambda\leq\bar{\lambda}_0$, $x\in\Sigma_{\lambda}$ and $|x|\geq R$.
\end{lemA}
\begin{lemA}\label{CGS-lem-2}\cite{CGS}
Suppose $v$ satisfies the assumption of Lemma \ref{CGS-lem-1}, and $v(x)>v(x^{\lambda}_0)$ for $x\in\Sigma_{\lambda_0}$. Assume $v(x)-v(x^{\lambda_0})$ is superharmonic in $\Sigma_{\lambda_0}$. Then there exist $\varepsilon>0$ and $S>0$ such that the followings hold.
\begin{itemize}
\item[(i)]
$v_{x_1}>0$ in $\{x:|x_1-\lambda_0|<\varepsilon,|x|>S\}$.
\item[(ii)]
$v(x)>v(x^{\lambda})$ in $\{x:x_1\geq\lambda_0|,|x|>S\}$.
\end{itemize}
for all$\lambda\leq\bar{\lambda}_0$, $x\in\Sigma_{\lambda}$ and $|x|\geq R$
for all $x\in\Sigma_{\lambda}$, $\lambda\leq\lambda_1$ with $|\lambda_1-\lambda_0|<c\varepsilon$, where $c_0$ is a small positive number depending on $\lambda_0$ and $v$ only.
\end{lemA}
\begin{proof}[\textbf{Proof of Theorem \ref{thm-classification}}]
Lemma \ref{alpha} and Lemma \ref{lem-u-asy} complete the proof of (i) and (ii) in Theorem (\ref{thm-classification}), respectively. Next we aim to prove the radial symmetry of solutions by the method of moving planes.
\textbf{Step 1:}
We start moving planes along $x_1$-direction. For any $\lambda$, we consider $w_{\lambda}(x)=u(x)-u(x^{\lambda})$ in $\Sigma_{\lambda}$. Then $w_{\lambda}(x)$ satisfies
\begin{equation}\label{equ-MP}
\left\{\begin{array}{lcl}
\Delta^2 w_{\lambda}(x)=b_{\lambda}(x)w_{\lambda}(x), && {\rm in} \ \; \Sigma_{\lambda},
\\
w_{\lambda}(x)=\Delta w_{\lambda}(x)=0, && {\rm on} \ \; \partial T_{\lambda},
\end{array}
\right.
\end{equation}
where
\begin{equation*}
b_{\lambda}(x)=6\frac{|x|^{4\gamma}e^{4u(x)}-|x^{\lambda}|^{4\gamma}e^{4u(x^{\lambda})}}{u(x)-u(x^{\lambda})}.
\end{equation*}
By (\ref{harmonic-asy}) and Lemma 2.3 in \cite{CGS}, we have $\Delta w_{\lambda}(x)=v(x^{\lambda})-v(x)<0$ for $x\in \Sigma_{\lambda}$, $\lambda<\bar{\lambda}_0$ and $|x|\geq R$. Since $v(x)>0$ in $\mathbb{R}^4$ and $v(x)\to 0$ as $|x|\to \infty$, there exists $\bar{\lambda}_1<\bar{\lambda}_0$ such that
\begin{equation*}
v(x^{\lambda})<v(x),\quad |x|\leq R,\ \, \lambda<\bar{\lambda}_1.
\end{equation*}
Therefore, we have
\begin{equation*}
\Delta w_{\lambda}(x)<0,\quad x\in \Sigma_{\lambda},\ \,\lambda<\bar{\lambda}_1.
\end{equation*}
By (\ref{u-asy}), $\lim\limits_{|x|\to +\infty}w_{\lambda}(x)=0$. Applying the maximum principle, we have
\begin{equation*}
w_{\lambda}(x)>0,\quad x\in \Sigma_{\lambda},\ \,\lambda<\bar{\lambda}_1.
\end{equation*}
Let
\begin{equation}\label{limit-location}
\lambda_0:=\sup\big\{\lambda:v(x^{\mu})<v(x)\quad{\rm for\ \,all}\ \,x\in\Sigma_{\lambda}\ \,{\rm and}\ \,\mu\leq\lambda\big\}
\end{equation}
Since $v(x)\to 0$ as $|x|\to\infty$, we can see that $\lambda_0<+\infty$. Next we claim
\begin{equation*}
u(x)\equiv u(x^{\lambda_0}),\quad {\rm for\ \, all\ \,}x\in\Sigma_{\lambda}.
\end{equation*}
Suppose $w_{\lambda_0}\not\equiv0$ in $\Sigma_{\lambda}$. By continuity, $\Delta w_{\lambda_0}(x)\leq 0$. Since $w_{\lambda_0}\to 0$ as $|x|\to+\infty$ and $w_{\lambda_0}\big|_{T_{\lambda_0}}=0$. By the strong maximum principle. we have
\begin{equation}\label{w-0}
w_{\lambda_0}(x)>0,\quad x\in \Sigma_{\lambda_0}.
\end{equation}
From (\ref{equ-liou-2}), we have
\begin{equation*}
\Delta^2 w_{\lambda_0}(x)=6\big(|x|^{4\gamma}e^{4u(x)}-|x^{\lambda_0}|^{4\gamma}e^{4u(x^{\lambda_0})}\big).
\end{equation*}
(\ref{w-0}) implies $u(x)>u(x^{\lambda_0})$ in $\Sigma_{\lambda_0}$. On the other hand, $|x|<|x^{\lambda_0}|$ and $\gamma<0$ imply $|x|^{4\gamma}>|x^{\lambda_0}|^{4\gamma}$. Hence
\begin{equation}\label{w-4}
\Delta^2 w_{\lambda_0}(x)>0,\quad x\in \Sigma_{\lambda_0},
\end{equation}
which means $\Delta w_{\lambda_0}$ is a subharmonic function. Applying the strong maximum principle again, we have
\begin{equation}\label{w-2}
\Delta w_{\lambda_0}(x)<0,\quad x\in \Sigma_{\lambda_0}.
\end{equation}
By the definition of $\lambda_0$, there exists $\lambda_n\nearrow\lambda_0$ and $x_n\in \Sigma_{\lambda_n}$ such that
\begin{equation*}
w_{\lambda_n}(x_n)=\sup_{\Sigma_{\lambda_n}}w_{\lambda_n}>0.
\end{equation*}
By Lemma 2.4 in \cite{CGS}, $\{x_n\}$ is bounded. Without loss of generality, we assume $x_0=\lim\limits_{n\to+\infty}x_n$.
If $x_0\in\Sigma_{\lambda_0}$, by continuity we have $\Delta w_{\lambda_0}(x_0)=0$, a contradiction to (\ref{w-2}).
If $x_0\in T_{\lambda_0}$, then $\nabla\big(\Delta w_{\lambda_0}(x_0)\big)=0$. However, by (\ref{w-4}) and (\ref{w-2}), $\Delta w_{\lambda_0}$ is a negative subharmonic function in $\Sigma_{\lambda_0}$. Thus by the Hopf boundary lemma, we have $\frac{\partial}{\partial\nu}\Delta w_{\lambda_0}(x_0)>0$, which yields a contradiction.
At this point, the claim is proved. Then we can change the $x_1$-direction to another one and finally obtain the radial symmetry.
\medskip
\textbf{Step 2:}
Now we prove the uniqueness of the solution of (\ref{equ-liou-2}) modulus scaling in (\ref{rescale}).
Let $w_1$ and $w_2$ be two radial solutions of (\ref{equ-liou-2}). Then $w_i$ satisfies $w_i^{'}(0)=w_i^{'}(0)=0$. Without loss of generality, we may assume $w_1(0)=w_2(0)$, on account of the rescale invariance. By the uniqueness of ODE, we only need to prove $w_1^{''}(0)=w_1^{''}(0)$.
If $w_1^{''}(0)<w_2^{''}(0)$, then $w_1(r)<w_2(r)$ for small $r>0$. We will prove $w_1(0)<w_2(0)$ for all $r>0$. Suppose there exists $r_0>0$ such that $w_1(r_0)=w_2(r_0)$ and $w_1(r)<w_2(r)$ for $0<r<r_0$. Then by (\ref{equ-liou-2}), we have
\begin{equation*}
\frac{\partial}{\partial r}\Delta (w_1(r)-w_2(r))=6r^{4\gamma}\big(e^{4w_1(r)}-e^{4w_2(r)}\big)<0,\quad 1<r\leq r_0,
\end{equation*}
which together with the assumption implies
\begin{equation*}
\Delta (w_1-w_2)<0,\quad {\rm in} \ \, B(0,r_0).
\end{equation*}
Since $w_1(r_0)-w_2(r_0)=0$, from the maximum principle, we have $w_1(r)-w_2(r)>0$ for $0<r< r_0$, which contradicts with $w_1(0)=w_2(0)$. Thus, $w_1(r)<w_2(r)$ for all $r>0$. Hence $\frac{\partial}{\partial r}\Delta (w_1(r)-w_2(r))<0$ for all $r>0$, which means $\Delta (w_1(r)-w_2(r))$ is decreasing in $r$. Thus $w_1(r)-w_2(r)\leq -c r^2$ as $r\to +\infty$ for some constant $c>0$, which yields a contradiction to the assumption $w_i(r)=o(r^2)$ at $\infty$.
Similarly, it is impossible for $w_1^{''}(0)>w_2^{''}(0)$. Thus, the radial solution of (\ref{equ-liou-2}) is unique under the rescale $u_{\lambda}(x)=u(\lambda x)+(1+\gamma)\log \lambda$ for some $\lambda>0$, and it is valid for (\ref{equ-liou}) after scaling.
\end{proof}
\medskip
Next, we consider the case without the assumption $|u(x)|=o(|x|^2)$ at $\infty$.
The following lemma is similar to Lemma 3.3 in \cite{lin-classification}. But we do not require $u$ is harmonic, and in fact $\Delta u\equiv const.$ is enough. Furthermore, when we apply this lemma in the proof of Theorem \ref{thm-classification-2}, it is uncertain that $w$ is harmonic.
\begin{lem}\label{lem-poly}
Suppose that $\Delta u =a$ in $\mathbb{R}^n$ for a constant $a\in \mathbb{R}$ such that $\exp(u-c|x|^2)\in L^1(\mathbb{R}^n)$ for some $c>0$. Then $u$ is a polynomial of order at most 2.
\end{lem}
\begin{proof}[\textbf{Proof}]
We only need to prove for any unit vector $\xi \in\mathbb{R}^n$, there holds
\begin{equation*}
u_{\xi\xi}(x)\equiv const.\quad {\rm in}\ \, \mathbb{R}^n.
\end{equation*}
By Liouville's Theorem, it suffices to prove $u_{\xi\xi}(x)$ is bounded from above by a constant independent of $x$. Without loss of generality, we may consider $x=0$ and $\xi=e_1$. For simplicity, we denote $B(0,r)$ and $\partial B(0,r)$ to $B_r$ and $\partial B_r$ respectively.
Since $\Delta u=a$, then $u_{x_1 x_1}$ is harnomic. Hence for any $r>0$, we have
\begin{equation}
u_{x_1x_1}(0)=\frac{1}{\sigma_n r^n}\int_{B_r}u_{x_1x_1}(x){\rm d}x=\frac{1}{\sigma_n r^n}\int_{\partial B_r}u_{x_1}\frac{x_1}{|x|}{\rm d}\sigma,
\end{equation}
where $\sigma_n$ is the volume of the unit ball in $\mathbb{R}^n$. Integrating the identity above from $0$ to $r$, we have
\begin{equation}\label{poly-0}
\begin{split}
\frac{\sigma_n}{n+1}r^{n+1}u_{x_1x_1}(0)=&\int_{ B_r}u_{x_1}\frac{x_1}{|x|}{\rm d}\sigma=-\int_{B_r}u\frac{\partial}{\partial x_1}\big(\frac{x_1}{|x|}\big){\rm d }x+\int_{\partial B_r}u\frac{x_1^2}{|x|^2}{\rm d}\sigma \\
=&-\int_{B_r}\frac{u}{|x|}{\rm d }x+\int_{B_r}u\frac{x_1^2}{|x|^3}{\rm d }x+\int_{\partial B_r}u\frac{x_1^2}{|x|^2}{\rm d}\sigma.
\end{split}
\end{equation}
Note that
\begin{equation*}
\int_{B_r}\frac{u}{|x|}{\rm d }x=n\sigma_n\int_{0}^{r}\Big(\dashint_{\partial B_s}u{\rm d}\sigma\Big)s^{n-2}{\rm d}s.
\end{equation*}
Since $\Delta u=a$, then
\begin{equation*}
\begin{split}
a\sigma_n s^n=&\int_{B_s}\Delta u(y){\rm d}y=\int_{\partial B_s}\frac{\partial u}{\partial \nu}{\rm d}\sigma=s^{n-1}\int_{|\omega|=1}\frac{\partial u}{\partial s}(s\omega){\rm d}\sigma_{\omega}\\
=&s^{n-1}\frac{\partial}{\partial s}\int_{|\omega|=1}u(s\omega){\rm d}\sigma_{\omega}.
\end{split}
\end{equation*}
That is
\begin{equation*}
\frac{\partial}{\partial s}\int_{|\omega|=1}u(s\omega){\rm d}\sigma_{\omega}=a\sigma_n s.
\end{equation*}
Integrating the identity above from $0$ to $s$, we obtain
\begin{equation*}
\int_{|\omega|=1}u(s\omega){\rm d}\sigma_{\omega}=\int_{|\omega|=1}u(0){\rm d}\sigma_{\omega}+\frac{a}{2}\sigma_n s^2,
\end{equation*}
which means
\begin{equation*}
\dashint_{\partial B_s}u{\rm d}\sigma=\frac{1}{n\sigma_n}\int_{|\omega|=1}u(s\omega){\rm d}\sigma_{\omega}=u(0)+\frac{a}{2n} s^2.
\end{equation*}
Then
\begin{equation}\label{poly-1}
\int_{B_r}\frac{u}{|x|}{\rm d }x=n\sigma_n\int_{0}^{r}\big(u(0)+\frac{a}{2n} s^2\big)s^{n-2}{\rm d}s=\frac{n}{n-1}\sigma_n r^{n-1}u(0)+\frac{a}{2(n+1)} r^{n+1}.
\end{equation}
On the other hand, by a direct computation, we have
\begin{equation}\label{poly-2}
\int_{B_r}u\frac{x_1^2}{|x|^3}{\rm d }x=\frac{\sigma_n}{n-1} r^{n-1}
+\int_{\partial B_r}u\frac{x_1^2}{|x|^2}{\rm d}\sigma
\end{equation}
and
\begin{equation}\label{poly-3}
\int_{\partial B_r}u\frac{x_1^2}{|x|^2}{\rm d}\sigma=\sigma_n r^{n-1}
\end{equation}
From (\ref{poly-0})$\sim$(\ref{poly-3}), we have
\begin{equation*}
\frac{r^2}{n+1}u_{x_1x_1}(0)=-\frac{n}{n-1}u(0)-\frac{a}{2(n+1)\sigma_n}r^2+\frac{1}{n-1}\dashint_{B_r}u{\rm d}\mu_1+\dashint_{\partial B_r}{\rm d}\mu_2,
\end{equation*}
where ${\rm d}\mu_1=\frac{x_1^2}{|x|^3}{\rm d }x$ and ${\rm d}\mu_2=\frac{x_1^2}{|x|^2}{\rm d }x$ on $\partial B_r$. By Jensen's inequality, we have
\begin{equation}\label{poly-4}
\begin{split}
&\exp\Big(\frac{r^2}{2(n+1)}u_{x_1x_1}(0)\Big) \\ \leq&\exp\Big(-\frac{n}{2(n-1)}u(0)\Big)\exp\Big(-\frac{a}{4(n+1)\sigma_n}r^2\Big)\Big(\dashint_{ B_r}e^{\frac{u}{2(n-1)}}{\rm d}\mu_1\Big)\Big(\dashint_{\partial B_r}e^{\frac{u}{2}}{\rm d}\mu_2\Big).
\end{split}
\end{equation}
For any $c_1>-\frac{a}{4(n+1)\sigma_n}$, denoting $\tilde{c}_1=c_1+\frac{a}{4(n+1)}$, we have
\begin{equation}\label{poly-5}
\begin{split}
&\int_{1}^{\infty}\exp\Big[\big(\frac{r^2}{2(n+1)}u_{x_1x_1}(0)-c_1\big)r^2\Big]{\rm d}r \leq\exp\Big(-\frac{n}{2(n-1)}u(0)\Big) \\
&\qquad\Big\{\int_{1}^{\infty}\big(\dashint_{B_r}e^{\frac{u}{n-1}}{\rm d}\mu_1\big)e^{-\tilde{c}_1r^2}{\rm d}r\Big\}^{\frac{1}{2}}\Big\{\int_{1}^{\infty}\big(\dashint_{\partial B_r}e^u{\rm d}\mu_2\big)e^{-\tilde{c}_1r^2}{\rm d}r\Big\}^{\frac{1}{2}}.
\end{split}
\end{equation}
By the assumption, we can choose $c_1$ large enough such that the right hand of (\ref{poly-5}) is finite. Therefore, we have
\begin{equation}\label{poly-6}
u_{x_1x_1}(0)\leq2(n+1)c_1.
\end{equation}
(\ref{poly-6}) is valid for all $x\in\mathbb{R}^n$. By Liouville's Theorem, we obtain $u_{x_1x_1}(x)\equiv const.$, which indicates $u$ is a polynomial of order at most 2.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{thm-classification-2}}]
Suppose that $u$ is a solution of (\ref{equ-liou-2}). Let $v$ be (\ref{v-def}) and $w(x)=u(x)+v(x)$. By Lemma \ref{lem-lap-u}, we have $\Delta w(x)\equiv-C_1\leq 0$ in $\mathbb{R}^4$. Then Lemma \ref{lem-poly} implies there exist constants $c_0$ and $a_{ij}$ $(i,j=1,\cdots,4)$ such that $a_{ij}=a_{ji}$ and
\begin{equation*}
w(x)=\sum_{i,j,k=1}^{4}(a_{ij}x_ix_j+b_kx_k)+c_0.
\end{equation*}
After an orthogonal transformation, we may assume
\begin{equation*}
u(x)=\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y-\sum_{j=1}^{4}(a_jx_j^2+b_jx_j)+c_0.
\end{equation*}
Since $|x|^{4\gamma}e^{4u}\in L^1(\mathbb{R}^4)$, we have $a_j\geq 0$ for all $j=1,\cdots,4$, and $b_j=0$ if $a_j=0$. Hence, we can rewrite $u(x)$ as follows:
\begin{equation*}
u(x)=\frac{3}{4\pi^2}\int_{\mathbb{R}^4}\log\big(\frac{|y|}{|x-y|}\big)|y|^{4\gamma}e^{4u(y)}{\rm d}y-\sum_{j=1}^{4}a_j(x_j-x_j^0)^2+c_0
\end{equation*}
and (\ref{lap-u-2}) holds from the previous argument.
\medskip
Next, we show the radial symmetry. Let $\hat{u}(x)=u(x)+\sum_{j=1}^{4}a_j(x_j-x_j^0)^2$. Then
\begin{equation}
\Delta^2\hat{u}(x)=6|x|^{4\gamma}e^{-4\sum_{j=1}^{4}a_j(x_j-x_j^0)^2}e^{4\hat{u}(x)},\quad{\rm in}\ \, \mathbb{R}^4.
\end{equation}
As in Lemma \ref{lem-u-asy}, we set $\hat{w}(x)=\hat{u}(\frac{x}{|x|^2})-\alpha\log|x|$, then $|\hat{w}(x)|=o(\log\frac{1}{|x|})$ near 0 from Lemma \ref{lem-v-upper} and Lemma \ref{lem-v-lower}, and $\hat{w}$ satisfies
\begin{equation}
\Delta^2\hat{w}(x)=6|x|^{4\gamma}e^{-4\sum_{j=1}^{4}a_j(\frac{x_j}{|x|^2}-x^0)^2}e^{4\hat{w}(x)},\quad{\rm in}\ \, \mathbb{R}^4\setminus\{0\}.
\end{equation}
Note that $a_j\geq 0$, then we can follow the argument in the proof of Lemma \ref{lem-u-asy} to obtain for $|x|$ large,
\begin{equation}
\hat{u}(x)=-\alpha\log|x|+c_0+O(\frac{1}{|x|}),
\end{equation}
and
\begin{equation}
\left\{\begin{array}{ll}
-\Delta \hat{u}(x)=\frac{2\alpha}{|x|^2}+O(\frac{1}{|x|^3}), \\
-\frac{\partial}{\partial x_i}\Delta \hat{u}(x)=-4\alpha\frac{x_i}{|x|^4}+O(\frac{1}{|x|^4}),\\
-\frac{\partial^2}{\partial x_i\partial x_j}\Delta \hat{u}(x)=O(\frac{1}{|x|^4}).
\end{array}
\right.
\end{equation}
At this point, we establish (\ref{rep-u-2}). Finally, as in the proof of Theorem \ref{thm-classification}, we can apply the method of moving planes to show that $\hat{u}(x)$ is symmetric with respect to the hyperplane $\{x:x_i=0\}$ if $a_i x_i^0= 0$. In particular, if $a_i x_i^0= 0$ for all $i=1,\cdots,4=0$, then $u$ is radially symmetric with respect to the origin.
\end{proof}
\section[preliminaries]{Preliminaries for blowup analysis}\label{preliminaries}
Let $G:M\times M\setminus {\rm diag}$ denote the Green's function for the Paneitz operator
\begin{equation}\label{Green-function}
f(x)-\bar{f}=\int_M G(x,y)P_g f(y){\rm d}V_g(y),\quad \int_M G(x,y){\rm d}V_g(y)=0,
\end{equation}
where $\bar{f}=\frac{1}{{\rm vol}_g(M)}\int_M f{\rm d }V_g$ is the average of $f$ over $M$. Then the weak form of (\ref{Green-function}) is
\begin{equation}\label{Green-weakequ}
P_{g,y}G(x,y)=\delta_x-\frac{1}{{\rm vol}_g(M)}.
\end{equation}
Set $R$ be the regular part of the Green function. Then by the Appendix A in \cite{zhang-weinstein}, for $y$ in a neighborhood of $x$,
\begin{equation}\label{Green-func-expression}
G(x,y)=-\frac{1}{8\pi^2}\log d_g(x,y) \chi+R(x,y).
\end{equation}
where $\chi$ is a cut-off function to avoid cut locus.
Using $G$ we can decompose $u_k$ as the sum of its regular part and singular part
\begin{equation}\label{decompose-uk}
u_k(x)=\tilde{u}_k(x)-8\pi^2\sum_{j=1}^{N}\gamma_j G(x,q_j).
\end{equation}
Then $\tilde{u}_k$ satisfies
\begin{equation}\label{equ-tilde-uk}
P_g\tilde{u}_k+2b_k=2H_ke^{4\tilde{u}_k}\quad {\rm in} \ \, M,
\end{equation}
where
\begin{equation}\label{Hk(x)}
H_k(x)=h_k(x)\prod_{j=1}^N e^{-32\pi^2\gamma_j G(x,q_j)}.
\end{equation}
Clearly, (\ref{Q-equation-blowup}) and (\ref{assumption-coe}) imply that
\begin{equation}\label{finite-integral-tildeuk}
\int_M H_ke^{4\tilde{u}_k}{\rm d}V_g\leq C
\end{equation}
We will work with $\tilde{u}_k$ in the later blow-up analysis.
Similar with \cite{zhang-weinstein}, since the metric $g$ may not be locally conformally flat, we will apply the {\itshape conformal normal coordinates}, whose existence has been proved in \cite{Lee-Parker}. More specially, for $q\in M$, there exists a normal coordinate around $q$ such that $g$ can be deformed to $\mathfrak{g}$ which satisfies $\det\,(\mathfrak{g})=1$. We use $R_{ijkl}$ to denote the curvature tensor under $\mathfrak{g}$.
We will apply the expansions of the metric $\mathfrak{g}$ and its derivatives in the conformal normal coordinates (seen in the Appendix B of \cite{zhang-weinstein}), which are
\begin{equation}\label{expansion-g}
\begin{split}
&\mathfrak{g}_{ab}(x)=\delta_{ab}+\frac{1}{3}R_{aijb}x^ix^j+O(r^3),\\
&\mathfrak{g}^{ab}(x)=\delta_{ab}-\frac{1}{3}R_{aijb}x^ix^j+O(r^3),\\
&\partial_c\mathfrak{g}^{ab}(x)=-\frac{1}{3}(R_{acib}+R_{aicb})x^i+O(r^2), \\
&\partial_{ab}\mathfrak{g}^{ab}(x)=\frac 13 R_{ia,a}x^i+O(r^2),
\end{split}
\end{equation}
In addition, the following Pohozaev identity from the Appendix D in \cite{zhang-weinstein} will play an important role when the blow-up analysis is carried out in the conformal normal coordinates.
\begin{lemA}\cite{zhang-weinstein}
For equation $P_gu+2b=2he^{4u}$ in $M$ and $\Omega=B(0,r)$, there holds
\begin{equation}\label{PI-mfd}
\begin{split}
&\int_{\Omega}\Big(2he^{4u}+\frac{1}{2}x^i\partial_i he^{4u}\Big) \\
=&\int_{\partial\Omega}\Big(\frac{1}{2}x^i\nu_ihe^{4u}-x^k\nu_jg^{ij}\partial_i(\Delta_g u)\partial_k u +\nu_jg^{ij}\Delta_gu\partial_iu +x^k\nu_jg^{ij}\Delta_gu \partial_{ik} u-\frac{1}{2}x^i\nu_i(\Delta_gu)^2 \Big) \\
&+\int_{\Omega}\Big(\Delta_gu\partial_i g^{ij}\partial_ju+x^k\Delta_gu\partial_{ik} g^{ij}\partial_ju+x^k\Delta_gu\partial_kg^{ij}\partial_{ij}u-2bx^i\partial_iu\Big) \\
&+2\int_{\partial\Omega}\Big(R_{ij,l}(q)x^lx^k\nu_i\partial_ju\partial_ku +O(r^3)|\nabla u|^2\Big) \\
&-\int_{\Omega}\Big(2R_{ij,l}(q)\big(x^l\partial_j u\partial_i u+x^kx^l\partial_j u\partial_{ik}u\big)+0(r^2)|\nabla u|^2+O(r^4)|\nabla^2 u|\Big)
\end{split}
\end{equation}
\end{lemA}
Note that we use $B(p,r)$ to denote a ball centered at $p$ with radius $r$. Sometimes if the center is the origin, we use $B_r$ instead of $B(0,r)$.
\section[Blow-up analysis]{Blow-up analysis near the singularity}\label{blowup-local}
In this section, we focus on the blow-up analysis near $q_j$, and to simplify the notation, we will omit the superscript $j$. Similar to the argument in \cite{zhang-weinstein}, we will work in the conformal normal coordinates near $q$ from \cite{Lee-Parker}. To be specific, we can find some function $w$ defined on $M$, such that in a small neighborhood $B(q,\delta)$ of $q$, $\delta>0$, we have
\begin{equation}
\det\,(\hat{g})=1
\end{equation}
in the normal coordinates of the conformal metric $\hat{g}=e^{2w}g$. For convenience we just use $g$ instead of $\hat g$. Note that in a neighborhood of $q$,
\begin{equation}\label{hat-w}
w(x)=O(d_g(x,q)^2).
\end{equation}
where $d_g(x,q)$ stands for the distance between $x$ and $q$ under metric $g$.
Using the conformal covariance property of $P_g$
the function $\mathfrak{u}_k=\tilde{u}_k-w$ satisfies
\begin{equation}\label{equ-hat-uk}
P_{g}\mathfrak{u}_k+2\mathfrak{b}_k=2H_ke^{4\mathfrak{u}_k},
\end{equation}
and
\begin{equation}\label{finite-int-hatuk}
\int_MH_ke^{4\mathfrak{u}_k}{\rm d}V_{g}\leq C,
\end{equation}
where $2\mathfrak{b}_k=P_{g}w+2b_k$ and $\tilde{H}_k=H_ke^{4w}$. For simplity, we still denote $\tilde{H}_k$ by $H_k$.
We still use $G$ to denote the Green's function for $P_{g}$. Then we have the following Green's representation formula
\begin{equation}\label{Green-formula}
\mathfrak{u}_k(x)=\bar{\mathfrak{u}}_k+2\int_MG(x,y)H_k(y)e^{4\mathfrak{u}_k(y)}{\rm d}V_{g}(y)-2\int_MG(x,y)\mathfrak{b}_k(y){\rm d}V_{g}(y),
\end{equation}
where $\bar{\mathfrak{u}}_k$ is the average of $\mathfrak{u}_k$ over $(M,g)$. Using the expression of $G$ in (\ref{Green-func-expression}, we have
\begin{equation}\label{Green-rep-formula}
\mathfrak{u}_k(x)=\bar{\mathfrak{u}}_k+2\int_M \Big(-\frac{1}{8\pi^2}\log d_g(x,y)\chi\Big) H_k(y)e^{4\mathfrak{u}_k(y)}{\rm d}V_{g}(y)+\mathfrak{\phi}_k(x),
\end{equation}
where
\begin{equation}\label{hat-phik}
\mathfrak{\phi}_k(x)=2\int_MR(x,y)H_k(y)e^{4\mathfrak{u}_k(y)}{\rm d}V_{g}(y)-2\int_MG(x,y)\mathfrak{b}_k(y){\rm d}V_{g}(y).
\end{equation}
Note that $\det\,(g)=1$ in $B(q,\delta)$, we have ${\rm d}V_{g}(y)={\rm d}y$ in $B(q,\delta)$. Taking the difference of (\ref{Green-rep-formula}) evaluated at $x$ and $q$, we get
\begin{equation}
\begin{split}
&\mathfrak{u}_k(x)-\mathfrak{u}_k(q)\\
=&\frac{1}{4\pi^2}\int_M\big(\chi(r_q)\log|y-q|-\chi(r_x)\log d_g(x,y)\big)H_k(y)e^{4\mathfrak{u}_k(y)}{\rm d}V_{g}(y)+\mathfrak{\phi}_k(x)-\mathfrak{\phi}_k(q),
\end{split}
\end{equation}
where $r_q=|y-q|$ and $r_x=d_g(x,y)$. Here since the coordinates are normal, we have $d_g(y,q)=|y-q|$.
Thanks to the cut-off function $\chi$, we can replace the integral over $M$ by an integral over $B(q,2\delta)$:
\begin{equation}\label{difference-Green-for}
\begin{split}
&\mathfrak{u}_k(x)-\mathfrak{u}_k(q)\\
=&\frac{1}{4\pi^2}\int_{B(q,4\delta)}\big(\chi(r_q)\log|y-q|-\chi(r_x)\log d_g(x,y)\big)H_k(y)e^{4\mathfrak{u}_k(y)}{\rm d}V_{g}(y)\\
&+\mathfrak{\phi}_k(x)-\mathfrak{\phi}_k(q),\quad x\in B(q,2\delta).
\end{split}
\end{equation}
\medskip
We next give an upper bound of the mass near $q$ when $\mathfrak{u}_k$ cannot blow up at $q$. Before stating such a small energy Lemma we point out that the function $H_k$ can be written in the neighborhood of a singular source $q$ as
\begin{equation}\label{eq:mathcalh}
H_k(x) = \mathfrak{h}_k(x)d_g(x,q)^{4\gamma},
\end{equation}
with $ \mathfrak{h}_k(q) \neq 0$. Using this notation, our result states as follows:
\begin{lem}\label{lem-small-mass-regular}
Let $q$ be a singular source with index $\gamma$. If
\begin{equation}\label{smallness-condition}
\int_{B(q,2\delta)}2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}{\rm d}V_{g}<\min\{8\pi^2,8\pi^2(1+\gamma)\},
\end{equation}
$\mathfrak{h}_k$ is defined in \eqref{eq:mathcalh}.
Then $\mathfrak{u}_k\leq C$ in $B(q,\delta)$.
\end{lem}
\begin{proof}
Note that
\begin{equation*}
P_{g}\mathfrak{u}_k=2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}-2\mathfrak{b}_k.
\end{equation*}
We write $\mathfrak{u}_k=u_{1k}+u_{2k}$ where $u_{1k}$ is the solution of
\begin{equation}\label{smallness-equ}
\left\{\begin{array}{lcl}
\Delta^2 u_{1k}=2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}&& {\rm in} \ \, B(q,2\delta)\\
u_{1k}(x)=\Delta u_{1k}(x)=0 && {\rm on} \ \, \partial B(q,2\delta).
\end{array}
\right.
\end{equation}
By Lemma \ref{lem-BM} from \cite{lin-classification} we have
\begin{equation}\label{BM-thm-1}
\int_{B(q,2\delta)}\exp\Big\{\frac{\tilde{\delta}|\mathfrak{u}_{1k}|}{\parallel 2H_ke^{4\mathfrak{u}_k} \parallel_{L^1(B(q,2\delta),g)}}\Big\}{\rm d}V_{g}\leq C,
\end{equation}
with any $\tilde{\delta}\in(0,32\pi^2)$ and some constant $C=C(\tilde{\delta},\delta)$. On one hand in $B(q,2\delta)$,
\begin{equation}\label{u1k}
u_{1k}(x)=\int_{B(q,\delta)}G_{\delta}(x,y)2\mathfrak{h}_kd_g(\eta, q)^{4\gamma}e^{4\mathfrak{u}_k}{\rm d}\eta, \quad x\in B(q,2\delta)
\end{equation}
where $G_{\delta}(x,y)$ is the Green's function of $\Delta^2$ on $B(q,2\delta)$:
$$G_{\delta}(x,y)=-\frac{1}{8\pi^2}\log |x-y|+R_{\delta}(x,y), $$
with
$$ G_{\delta}(x,y)=\Delta_yG_{\delta}(x,y)=0,\quad \mbox{for } x\in B(q,2\delta), y\in \partial B(q,2\delta). $$
In particular for $x\in B(q, \frac 32\delta)$,
$$u_{1k}(x)=-\frac 1{8\pi^2}\int_{B(q,2\delta)}\log |x-q|2\mathfrak{h}_k|\eta -q|^{4\gamma}e^{4\mathfrak{u}_k}{\rm d}\eta+O(1),\quad x\in B(q,\frac 32\delta). $$
On the other hand the Green's representation formula of $\mathfrak{u}_k$ gives
$$\mathfrak{u}_k(x)=u_{1k}(x)+u_{2k}(x)=\overline{ \mathfrak{u}_k}+\int_MG(x,\eta)2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k}{\rm d}\eta. $$
Since the leading term of $G$ and $G_{\delta}$ are both $-\frac 1{8\pi^2}\log d_g(x,q)$, we have
$$u_{2k}(x)=\overline{\mathfrak{u}_k}+O(1). $$
From $\int_Me^{4\mathfrak{u}_k}{\rm d}V_g\le C$ and Jensen's inequality
$$e^{4\bar{\mathfrak{u}}_k}\le \int_M e^{4\mathfrak{u}_k}{\rm d}V_g\le C.$$
Therefore, $\mathfrak{u}_{2k}\leq C$ in $B(q,\frac{3}{2}\delta)$. Now we focus on $u_{1k}$.
If $\gamma>0$, (\ref{smallness-condition}) is
\begin{equation*}
\int_{B(q,2\delta)}2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}{\rm d}V_{g}<8\pi^2.
\end{equation*}
Since $u_{2k}$ is bounded from above in $B(q,\frac 32\delta)$, we see from
(\ref{BM-thm-1}) that there exists some $p>1$ to make $e^{4u_{1k}}\in L^{p}(B(q,2\delta))$:
\begin{equation}\label{Lp'-estimate}
\parallel 2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}\parallel_{ L^{p}(B(q,\frac{3}{2}\delta))}\leq C.
\end{equation}
The estimate (\ref{Lp'-estimate}) leads to a $L^{\infty}$ of $u_{1k}$ in $B(q,\delta)$ based on two reasons. First
the integration of (\ref{u1k}) gives a $L^1$ bound of $u_{1k}$ in $B(q,\frac 32\delta)$:
\begin{equation}
\parallel u_{1k} \parallel_{L^1(B(q,\frac{3}{2}\delta))}\leq C.
\end{equation}
Second, the standard interior regularity results in \cite{Browder-regularity} (Theorem 1 in Section 3 of \cite{Browder-regularity}) gives
\begin{equation*}
\parallel u_{1k}\parallel_{W^{4,p}B(q,\delta)}\leq \parallel 2\mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}\parallel_{ L^{p}(B(q,\frac{3}{2}\delta))}+\parallel u_{1k} \parallel_{L^1(B(q,\frac{3}{2}\delta))}\leq C.
\end{equation*}
Thus we have obtained the $L^{\infty}$ bound of $u_{1k}$ in $B(q,\delta)$ by standard Sobolev embedding theorem.
If $-1<\gamma<0$, (\ref{smallness-condition}) is
\begin{equation*}
\int_{B(q,2\delta)}2\mathfrak{h}_kd_g(x,q)^{-4|\gamma |}e^{4\mathfrak{u}_k(x)}{\rm d}V_{g}<8\pi^2(1-|\gamma |).
\end{equation*}
Since $\frac{\tilde \delta}{8\pi^2(1-|\gamma |)}<\frac{4}{1-|\gamma|}$, this strict inequality makes it possible to choose $\tilde{\delta}\in(0,32\pi^2)$ and $p>1$ with
\begin{equation*}
p<\frac{1}{|\gamma |}\quad p<\frac{\tilde{\delta}}{32\pi^2(1-|\gamma |)}.
\end{equation*}
Then H${\rm \ddot{o}}$lder inequality tells us that (\ref{Lp'-estimate}) is also true in this case and the $L^{\infty}$ bound of $u_{1k}$ over $B(q,\delta)$ follows immediately. The combination of the $L^{\infty}$ bound of $u_{1k}$ and the upper bound of $u_{2k}$ implies that $\mathfrak{u}_k\le C$ in $B(q, \delta)$.
\end{proof}
\begin{rem} \label{rem-small} If $q$ is a regular point the same proof obviously leads to a similar conclusion: Suppose $B(q,2\delta)$ has no singular source and
$\int_{B(q,2\delta)}2\mathfrak{h}_k e^{4\mathfrak{u}_k}<8\pi^2$, there is an upper bound of $\mathfrak{u}_k$ in $B(q,\delta)$.
\end{rem}
Another immediate consequence of Lemma \ref{lem-small-mass-regular} is that blowup sequence only converges to point measures. The following theorem takes one step further to assert that the point measure is quantized and the bubbling solutions tend to $-\infty$ away from blowup points.
\begin{thm}[Concentration and Quantization]\label{thm-con-quan}
Let $\{\mathfrak{u}_k\}$ be a sequence of solution to (\ref{equ-hat-uk}) with (\ref{finite-int-hatuk}). Assume that $q$ is the only blow-up point of $\mathfrak{u}_k$ in $B(q,\delta)$, then as $k\to+\infty$, along a subsequence, there hold
\begin{equation}\label{con-neg-infty}
\mathfrak{u}_k\to-\infty,\quad uniformly\ \, on \ \, any\ \, compact \ \, set\ \, of\ \, B(q,\delta)\setminus\{q\},
\end{equation}
\begin{equation}\label{con-measure}
2 \mathfrak{h}_kd_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k}\to\beta\delta_q,\quad weakly\ \, in \ \, the\ \, sense \ \, of\ \,measure\ \, on\ \,M,
\end{equation}
with $\beta=16\pi^2(1+\gamma)$ and $\mathfrak{h}_k$ is defined in \eqref{eq:mathcalh}.
In particular, along a subsequence,
\begin{equation}\label{con-mass}
2 \int_{B(q,\delta)}H_k(x)e^{4\mathfrak{u}_k(x)}{\rm d}V_{g}\to16\pi^2(1+\gamma),\quad as\ \, k\to+\infty.
\end{equation}
\end{thm}
\begin{proof}[\textbf{Proof}]
Suppose $q$ is the only blowup pint in $B(q,2\delta)$, then we observe that for any given $K\subset\subset B(q,\delta)$
\begin{equation}\label{bd-osi}
|\mathfrak{u}_k(x)-\mathfrak{u}_k(y)|\le C(K)
\end{equation}
because the $|x-q|$ is comparable to $|y-q|$ and the total integration of $\mathfrak{h}_ke^{4\mathfrak{u}_k}$ is bounded. It is easy to obtain (\ref{bd-osi}) from the Green's representation formula. Then it is also immediate to observe that
\begin{equation}\label{uk-grad}
|\nabla_{g}^j\mathfrak{u}_k(x)|_{g}\leq C(K),\quad {\rm in}\ \,K,\quad j=1,2,3
\end{equation}
because when $x\in K$,
$$|\nabla^jG(x,y)|\le C|x-y|^{-j}, \quad x,y\in B(q,2\delta), $$
and $\mathfrak{u}_k$ has a upper bound. These two facts imply (\ref{uk-grad} by trivial estimates. Then the equation for $\mathfrak{u}_k$ further provides the estimate for the fourth order derivatives of $\mathfrak{u}_k$:
$$|\nabla^{\alpha}\mathfrak{u}(x)|\le C(k), \quad x\in K, \quad |\alpha|=4. $$
Now we prove (\ref{con-neg-infty}) by contradiction. Suppose that there exists a point $x_0\in B(q,\delta)\setminus\{q\}$ such that $\{\mathfrak{u}_k(x_0)\}_{k\in\mathbb{N}}$ is also bounded from below. By (\ref{bd-osi}) we see that $\mathfrak{u}_k$ is bounded in $L^{\infty}$ norm in any compact subset of $B(q,\delta)\setminus\{q\}$. This fact and the gradients estimates of $\mathfrak{u}_k$ guarantee that along a subsequence
\begin{equation*}
\mathfrak{u}_k\to \mathfrak{u}_0\quad {\rm in}\ \,C_{loc}^{3,\sigma}(B(q,2\delta)\setminus\{q\}),
\end{equation*}
with some constant $\sigma\in(0,1)$ and the limit function $\mathfrak{u}_0$ solves
\begin{equation*}
P_{g}\mathfrak{u}_0(x)+2\mathfrak{b}_0(x)=2\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0(x)} \quad {\rm in} \ \,B(q,2\delta)\setminus\{q\}.
\end{equation*}
Around $q$ we use $\beta(r)$ and its limit to describe the concentration of energy:
\begin{equation*}
\begin{split}
&\beta_k(r)=\int_{B(q,r)}2\mathfrak{h}_k(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}{\rm d}V_{g} \\
&\beta(r)=\lim_{k\to+\infty}\beta_k(r),\quad \beta=\lim_{r\to 0}\beta(r).
\end{split}
\end{equation*}
From Lemma \ref{lem-small-mass-regular} and Remark \ref{rem-small} we see that if $q$ is singular point $\beta\ge \min\{8\pi^2(1+\gamma),8\pi^2\}$, if $q$ is a regular point $\beta\ge 8\pi^2$.
Fixing any $r>0$ small, we integrate the equation of $\mathfrak{u}_k$ in $B(q,r)$ to obtain
\begin{equation*}
\int_{B(q,r)}\big(P_{g}\mathfrak{u}_k+2\mathfrak{b}_k\big){\rm d}V_{g}=\int_{B(q,r)}H_ke^{4\mathfrak{u}_k}{\rm d}V_{g}=\beta_k(r),
\end{equation*}
which implies
\begin{equation*}
\lim_{r\to 0}\int_{B(q,r)}\big(P_{g}\mathfrak{u}_0+2\mathfrak{b}_0\big){\rm d}V_{g}=\beta.
\end{equation*}
Therefore, $\mathfrak{u}_0$ satisfies, in the distribution sense,
\begin{equation*}
P_{g}\mathfrak{u}_0(x)+2\hat{b}_0(x)=2\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0(x)}+\beta \delta_q \quad {\rm in} \ \,B(q,2\delta)
\end{equation*}
Using the Green's representation formula for $\mathfrak{u}_k$, we have
\begin{equation}\label{hatu0-decompose}
\mathfrak{u}_0(x)=-\frac{\beta}{8\pi^2}\log d_g(x,q)+v(x)+w(x)
\end{equation}
where the first term comes from the convolution of $-\frac 1{8\pi^2}\log d_g(x,y)\chi$ with $\beta \delta_q$, the second term $v$ comes from
the convolution of $-\frac 1{8\pi^2}\log d_g(x,y)\chi $ with $2\mathfrak{h}_0d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0}$:
\begin{equation}\label{hatu0-v}
v(x)=-\frac{1}{4\pi^2}\int_{B(q,\delta)}\log d_g(x,y)\mathfrak{h}_0(y)d_g(y,q)^{4\gamma}e^{4\mathfrak{u}_0(y)}{\rm d}V_{g}(y),
\end{equation}
and $w$ is the collection of insignificant other terms:
\begin{equation}\label{hatu0-w}
w\in C^4(M).
\end{equation}
For $v$ we use (\ref{hatu0-v}) to denote $v(x)$ in $B(q,\delta)$ and we extend $v$ to make $v\equiv 0$ on $M\setminus B(q,2\delta)$.
Based on the definition of $v$ we now show that $v\in L^{\infty}(B(q,\delta))$. In fact, from (\ref{hatu0-v}) we have this lower bound of $v$ in $B(q,\delta)$:
\begin{equation}\label{v-lb}
v(x)\geq \frac{1}{4\pi^2}\log\frac{1}{\delta}\parallel V\parallel_{L^1(M)}\geq C\quad {\rm in} \ \, B(q,\delta)
\end{equation}
where
$$V(x)=\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0(x)}. $$
The lower bounds of $\mathfrak{h}_0$ and $v$ lead to a lower bound for $V(x)$:
\begin{equation*}
V(x)=\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0(x)}\geq Cd_g(x,q)^{4\gamma-\frac{\beta}{2\pi^2}}e^{4v(x)+w(x)}\geq\frac{c}{d_g(x,q)^s}
\end{equation*}
with $s=\frac{\beta}{2\pi^2}-4\gamma$ and suitable $c>0$. Since $\|V\|_{L^1}<\infty$ we see immediately that $s<4$, which is
\begin{equation}\label{beta-con-1}
\beta<8\pi^2(1+\gamma).
\end{equation}
Thus there is no way for $\mathfrak{u}_k$ to be bounded from below away from singular source unless $\gamma>0$. We have proved (\ref{con-neg-infty}) for $\gamma\le 0$.
For $\gamma>0$ we have an upper bound for $V(x)$:
\begin{equation}
V(x)\leq\frac{c}{d_g(x,q)^s}e^{4v(x)}\quad {\rm in} \ \, B(q,\delta), \quad \mbox{if}\quad \gamma>0.
\end{equation}
To proceed with the proof of $v\in L^{\infty}$ we observe from (\ref{hatu0-v}) and direct computation that
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta^2 v(x)=V(x)+\eta(x)&& {\rm in} \ \, B(q,2\delta)\\
v(x)=\Delta v(x)=0 && {\rm on} \ \, \partial B(q,2\delta),
\end{array}
\right.
\end{equation*}
where $\eta$ is smooth in $B(q,2\delta)$. Note that the boundary condition of $v$ on $\partial B(q,2\delta)$ is based on the smooth extension of $v$ mentioned before. Now we employ a standard argument of Brezis-Merle \cite{BM} to obtain $e^{\kappa |v|}\in L^1(B(q,2\delta),g)$ for any constant $\kappa>0$. Indeed, let $0<\varepsilon<1/\kappa$ and $V=V_1+V_2$ with $\parallel V_1\parallel_{L^1(B(q,2\delta))}<\epsilon$ and $V_2\in L^{\infty}(B(q,2\delta))$. Correspondingly we write $v=v_1+v_2$, where $v_1$ solves
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta^2 v_1(x)=V_1(x)&& {\rm in} \ \, B(q,2\delta)\\
v_1(x)=\Delta v_1(x)=0 && {\rm on} \ \, \partial B(q,2\delta),
\end{array}
\right.
\end{equation*}
and $v_2$ solves
\begin{equation*}
\left\{\begin{array}{lcl}
\Delta^2 v_2(x)=V_2(x)+\eta(x)&& {\rm in} \ \, B(q,2\delta)\\
v_2(x)=\Delta v_2(x)=0 && {\rm on} \ \, \partial B(q,2\delta),
\end{array}
\right.
\end{equation*}
Choosing $\tilde{\delta}=32\pi^2-1$ in (\ref{BM-thm-1}), we find $\int_{B(q,2\delta)}e^{\kappa |v_1|}\leq\int_{B(q,2\delta)}e^{\frac{|v_1|}{\parallel V_1\parallel_{L^1(B(q,2\delta))}}}\leq C$ based on the smallness of $\epsilon$. Then by standard elliptic regularity theory, we have $v_2\in L^{\infty}(B(q,2\delta))$. Consequently, $e^{\kappa |v|}\in L^1(B(q,2\delta),g)$. Since $\kappa$ is large it is possible to use H${\rm \ddot{o}}$lder inequality to
obtain $V\in L^{p'}(B(q,2\delta),\hat{g})$ for some $p\in(1,\frac{4}{s})$ if $s\geq 0$ and $p\in(1,+\infty)$ if $s\leq 0$. Thus we have proved $v\leq C$ in $B(q,2\delta)$.
From the $L^{\infty}$ bound of $v$ we can use two positive constants $c_1$ and $c_2$ to bound $V$ from above and below
\begin{equation}\label{range-V}
\frac{c_1}{d_g(x,q)^s}\leq V(x)=\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0(x)}\leq \frac{c_2}{d_g(x,q)^s},\quad \mbox{if}\quad \gamma>0.
\end{equation}
Next, we aim to derive a contradiction by taking advantage of the Pohozaev identity (\ref{PI-mfd}) in Section \ref{preliminaries}.
Set $h(x)=\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}$ and $\Omega=B(q,r)$ in (\ref{PI-mfd}), then direct computation and (\ref{comparsion-dist}) give rise to
\begin{equation*}
\int_{\Omega}x^i\partial_i\big(\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}\big)e^{4\mathfrak{u}_0}=\int_{B(q,r)}\big(x^i\partial_i \mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0}+x^i\mathfrak{h}_0\partial_i d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0}\big)
\end{equation*}
and
\begin{equation*}
\begin{split}
\partial_i d_g(x,q)^{4\gamma}=&4\gamma d_g(x,q)^{4\gamma-1}\partial_i d_g(x,q)=4\gamma d_g(x,q)^{4\gamma-1}\big(\partial_i|x-q|+C\big)\\
=&4\gamma d_g(x,q)^{4\gamma-2}\big((x-q)^i+C\big)
\end{split}
\end{equation*}
Immediately, together with (\ref{comparsion-dist}) we get
\begin{equation*}
\begin{split}
&\int_{\Omega}\frac{1}{2}(x-q)^i\partial_i\big(\mathfrak{h}_0(x)d_g(x,q)^{4\gamma}\big)e^{4\mathfrak{u}_0}\\
=&\int_{B(q,r)}2\gamma \mathfrak{h}_0(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0}+\int_{B(q,r)}\frac{1}{2}\partial_{\nu} \mathfrak{h}_0 d_g(x,q)^{4\gamma+1}e^{4\mathfrak{u}_0}
\end{split}
\end{equation*}
Therefore, for $r\to 0$
\begin{equation}\label{LHS-PI}
{\rm (LHS)\ \,of\ \,}(\ref{PI-mfd})=2(1+\gamma)\int_{B(q,r)}\mathfrak{h}_0d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_0}+O(r)=(1+\gamma)\beta+o_r(1),
\end{equation}
where $\lim_{r\to 0}o_r(1)=0$. Denote the four integrals on the right hand side of (\ref{PI-mfd}) by $I_1$, $I_2$, $I_3$ and $I_4$, respectively.
Thanks to the expansions of $g$, which are in (\ref{expansion-g}), we obtain that
\begin{equation*}
|I_2|\leq C\int_{B(q,r)}\Big(r|\nabla^2\mathfrak{u}_0||\nabla\mathfrak{u}_0|+|x-q||\nabla^2\mathfrak{u}_0||\nabla\mathfrak{u}_0|+r|x-q||\nabla^2\mathfrak{u}_0|+|x-q||\nabla \mathfrak{u}_0|\Big),
\end{equation*}
\begin{equation*}
|I_3|\leq C\int_{\partial B(q,r)}\Big(|x-q|^2|\nabla\mathfrak{u}_0|^2+O(r^3)|\nabla\mathfrak{u}_0|^2\Big),
\end{equation*}
\begin{equation*}
|I_4|\leq C\int_{B(q,r)}\Big(|x-q||\nabla\mathfrak{u}_0|^2+|x-q|^2|\nabla^2\mathfrak{u}_0||\nabla\mathfrak{u}_0|+O(r^2)|\nabla\mathfrak{u}_0|^2+O(r^4)|\nabla^2\mathfrak{u}_0|\Big),
\end{equation*}
Next, we shall estimate $|\nabla^j\mathfrak{u}_0|$ in $B(q,r)$ for $j=1,2,3$. Recalling (\ref{hatu0-decompose})$\sim$(\ref{hatu0-w}), it is important to consider the $\nabla^j v$ in $B(q,r)$ for $j=1,2,3$.By means of the Green's representation formula, we observe that
\begin{equation*}
\begin{split}
|\nabla^jv(x)|\leq C\int_{B(q,2\delta)}\frac{1}{d_g(x,y)^j}V(y)+O(1)
\end{split}
\end{equation*}
In order to estimate the integral in the inequality above, we decompose $B(q,2\delta)$ into two parts
\begin{equation*}
\Omega_1=B(q,2\delta)\cap\Big\{d_g(x,y)\leq\frac{d_g(x,q)}{2}\Big\},\quad \Omega_2=B(q,2\delta)\setminus \Omega_1.
\end{equation*}
In this estimate we use $V(y)=O(1)d_g(y,q)^{-s}$ in (\ref{range-V}). Hence
\begin{equation}\label{inte-omega1}
\int_{\Omega_1}\frac{1}{d_g(x,y)^j}V(y){\rm d}V_{g}(y)\leq \frac{C}{d_g(x,q)^s}\int_{B(x,\frac{d_g(x,q)}{2})}\frac{1}{d_g(x,y)^j}{\rm d}V_{g}(y)\leq C d_g(x,q)^{4-s-j}.
\end{equation}
Using (\ref{range-V}) again, we obtain that
\begin{equation*}
\begin{split}
\tilde{I}:=\int_{\Omega_2}\frac{1}{d_g(x,y)^j}V(y){\rm d}V_{g}(y)\leq C\int_{\Omega_2}\frac{1}{d_g(x,y)^j}\frac{1}{d_g(y,q)^s}{\rm d}V_{g}(y).
\end{split}
\end{equation*}
Fixing some $t$ as follows
\begin{equation}\label{t-range}
t\in \left\{\begin{array}{lcl}
(0,\frac{4}{s}),&& s>0,\\
(1,+\infty),&& s\leq 0,
\end{array}
\right.
\end{equation}
we have $-ts>-4$. It follows from the H$\ddot{\rm o}$lder inequality that
\begin{equation*}
\begin{split}
\tilde{I}\leq C\Big(\int_{\Omega_2}\frac{1}{d_g(x,y)^{jt^*}}{\rm d}V_{g}(y)\Big)^{\frac{1}{t^*}}\Big(\int_{\Omega_1}\frac{1}{d_g(y,q)^{st}}{\rm d}V_{g}(y)\Big)^{\frac{1}{t}}
\leq C\Big(\int^{\tilde{c}}_{\frac{d_g(x,q)}{2}}\frac{1}{\rho^{jt^*-3}}{\rm d}\rho\Big)^{\frac{1}{t^*}}
\end{split}
\end{equation*}
where $t^*=\frac{t}{t-1}$ denotes the conjugate of $t$ and $\tilde{c}$ is some positive constant. Then direct computation and the fact $-ts>-4$ imply that
\begin{equation}\label{inte-omega2}
\tilde{I}=\int_{\Omega_2}\frac{1}{d_g(x,y)^j}V(y){\rm d}V_{g}(y)\leq
\left\{\begin{array}{lcl}
C|\log d_g(x,q)|^{\frac{1}{t^*}},&& {\rm if} \ \,jt^*=4,\\
Cd_g(x,q)^{\frac{4}{t^*}-j}+C,&& {\rm if} \ \,jt^*\neq 4.
\end{array}
\right.
\end{equation}
In view of (\ref{t-range}), we get that
\begin{equation*}
t^*\in \left\{\begin{array}{lcl}
(\frac{4}{4-s},+\infty),&& s>0,\\
(1,+\infty),&& s\leq 0.
\end{array}
\right.
\end{equation*}
Hence, there holds $\frac{4}{t^*}-j<4-s-j$. Consequently, from (\ref{inte-omega1}) and (\ref{inte-omega2}) there exists some $\tau>0$ such that for any $r\in(0,\delta)$
\begin{equation}\label{est-dv-inball}
|\nabla^j v(x)|\leq Cd_g(x,q)^{\tau-j}+C,\quad j=1,2,3,\quad x\in B(q,r).
\end{equation}
In fact, we may choose $\tau\in(0,1)$ if $jt^*=4$, and otherwise $\tau=\frac{4}{t^*}$. At this point, we obtain that
\begin{equation}
|\nabla^j \mathfrak{u}_0(x)|\leq Cd_g(x,q)^{-j},\quad j=1,2,3\quad x\in B(q,r).
\end{equation}
Thus by virtue of (\ref{comparsion-dist}) and (\ref{est-dv-inball}), we may adjust $\tau>0$ such that on $\partial B(q,r)$
\begin{equation*}
\begin{split}
&\frac{\partial\mathfrak{u}_0}{\partial r}=-\frac{\beta}{8\pi^2}\frac{1}{|x-q|}+O(r^\tau)\frac{1}{|x-q|}+O(1), \\
&\Delta\mathfrak{u}_0=-\frac{\beta}{4\pi^2}\frac{1}{|x-q|^2}+O(r^\tau)\frac{1}{|x-q|^2}+O(1), \\
&\frac{\partial\Delta\mathfrak{u}_0}{\partial r}=\frac{\beta}{2\pi^2}\frac{1}{|x-q|^3}+O(r^\tau)\frac{1}{|x-q|^3}+O(1).
\end{split}
\end{equation*}
\medskip
Therefore, the estimates of $I_2$, $I_3$ and $I_4$ can be improved:
\begin{equation*}
\begin{split}
|I_2|&\leq C\int_{B(q,r)}\big(rd^{-3}+d^{-2}+1\big){\rm d}V_{g}\leq Cr^2, \\
|I_3|&\leq C\int_{\partial B(q,r)}r^2d_g(x,q)^{-2}{\rm d}V_{g}\leq Cr^3, \\
|I_4|&\leq C\int_{B(q,r)}d_g(x,q)^{-1}{\rm d}V_{g}\leq Cr^3. \\
\end{split}
\end{equation*}
Finally for $I_1$ and we use the expansions of $g^{ij}$ to obtain
\begin{equation*}
\begin{split}
I_1=&\int_{\partial B(q,r)}\big(-r\nu_i\partial_i(\Delta_{g}\mathfrak{u}_0)\partial_{\nu}\mathfrak{u}_0+\Delta_{g}\mathfrak{u}_0\partial_{\nu}\mathfrak{u}_0+(x-q)^k\nu_i\Delta_{g}\mathfrak{u}_0\partial_{ik}\mathfrak{u}_0-\frac{1}{2}r(\Delta_{g}\mathfrak{u}_0)^2\big)\\
&+O(r)\\
=&\int_{\partial B(q,r)}\Big(-r\nu_i\partial_i(\Delta\mathfrak{u}_0)\partial_{\nu}<x-q,\mathfrak{u}_0>_{flat}+\Delta\mathfrak{u}_0\partial_{\nu}\mathfrak{u}_0-\frac{1}{2}r(\Delta\mathfrak{u}_0)^2\Big)+O(r),
\end{split}
\end{equation*}
where we have used $\det\,(g)=1$ in $B(q,\delta)$ and $\Delta_{g}u=\partial_i(g^{ij}\partial_j u)$. Consequently,
\begin{equation}\label{RHS-PI}
{\rm (RHS)\ \,of\ \,}(\ref{PI-mfd})=I_1+o_r(1)=\frac{\beta^2}{16\pi^2}+o_r(1).
\end{equation}
Combining (\ref{LHS-PI}) and (\ref{RHS-PI}), we derive that $\beta=16\pi^2(1+\gamma)$, which yields a contradiction to (\ref{beta-con-1}) in the case $\gamma>0$. Therefore $\mathfrak{u}_k\to-\infty$ uniformly on any compact subset of $B(q,2\delta)\setminus\{q\}$, $\mathfrak{h}_k(x)d_g(x,q)e^{4\mathfrak{u}_k}(x)\to 0$ uniformly on any compact subset of $B(q,2\delta)\setminus\{q\}$ and
\begin{equation*}
\mathfrak{h}_k(x)d_g(x,q)e^{4\mathfrak{u}_k}(x)\to \beta\delta_q \ \,{\rm\ \,weakly\ \,in\ \,the\ \,sense\ \,of\ \,measure\ \,on\ \,}M.
\end{equation*}
In the end, we show the quantization $\beta$ is exactly $16\pi^2(1+\gamma)$. To see this, set $c_k=\dashint\mathfrak{u}_k{\rm d}\sigma$ and $\check{u}_k(x)=\mathfrak{u}_k(x)-c_k$. Then, we have $c_k\to-\infty$ and $\check{u}_k\to\check{u}$ in $C^4(B(q,\delta))$ as $k\to+\infty$. Moreover, there exists a smooth function $\check{v}$ such that $\check{u}(x)=\frac{\beta}{8\pi^2}\log d_g(x,q)+\check{v}$. Taking advantage of the Pohozaev identity as before, we obtain $\beta=16\pi^2(1+\gamma)$ and
\begin{equation*}
2\mathfrak{h}_k(x)d_g(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}\to 16\pi^2(1+\gamma)\delta_q.
\end{equation*}
\end{proof}
Based on Theorem \ref{thm-con-quan} and its proof, we immediately obtain the following corollaries.
\begin{cor}
Suppose that $\mathfrak{u}_k$ satisfies the assumptions in Theorem \ref{thm-con-quan}, then along a subsequence, there holds
\begin{equation*}
\mathfrak{u}_k-c_k\to-2(1+\gamma)\log d_g(x,q)+\hat{v},\quad in \ \,C^4_{loc}(B(q,2\delta)\setminus\{q\}),
\end{equation*}
where $c_k=\dashint_{\partial B(q,\delta)}\mathfrak{u}_k{\rm d}\sigma_{g}\to-\infty$ and $\hat{v}$ is a smooth function in $B(q,2\delta)$.
\end{cor}
\begin{cor}
Suppose that $\mathfrak{u}_k$ satisfy
$$P_{g} \mathfrak{u}_k(x)+2\mathfrak{b}_k=2H_ke^{4\mathfrak{u}_k}\quad in \ \, M$$
with $\int_{B(q_j,2\delta)}2H_ke^{4\mathfrak{u}_k}{\rm d}V_{g}\to \rho_j<16\pi^2(1+\gamma_j)$ for some $j\in\{1,\cdots,N\}$. Then $\{\mathfrak{u}_k\}$ is uniformly bounded from above on any subset of $B(q_j,2\delta)$. In particular, $\{\mathfrak{u}_k\}$ can not blow up in $B(q_j,2\delta)$.
\end{cor}
\section[A Priori Estimate]{Concentration-Compactness Result and A Priori Estimate}\label{CC-Apriori}
In this section, we aim to establish the concentration-compactness principle and a priori estimate basing on the result in the section \ref{blowup-local}. Indeed, we will derive the following concentration-compactness type result for the regular part of $\{u_k\}$.
Set
\begin{equation*}
\rho_k=\int_M 2H_ke^{4\mathfrak{u}_k}{\rm d}V_{\mathfrak{u}_k}.
\end{equation*}
\begin{thm}[Concentration-Compactness]\label{thm-concentration-compactness}
Let $\{\tilde{u}_k\}$ be a sequence of solution to (\ref{equ-tilde-uk}) and (\ref{finite-integral-tildeuk}) with $\rho_k\to\rho$. Then there exists a subsequence, still denoted $\{\tilde{u}_k\}$, for which one of the following alternative holds:
\begin{itemize}
\item[(i)]
$\sup_{\varSigma}|\tilde{u}_k|\leq C_{\varSigma}$, for any $\varSigma\subset\subset M$.
\item[(ii)]
$\sup_{\varSigma}\tilde{u}_k\to-\infty$, for any $\varSigma\subset\subset M$.
\item[(iii)]
There exist a finite set $S=\{p^1,\cdots,p^m\}\subset M$ with $m\in \mathbb{N}$, and sequences of points $\{x_k^1\}_{k\in\mathbb{N}},\cdots,\{x_k^m\}_{k\in\mathbb{N}}\subset M$, such that for all $i=1,\cdots,m$
\begin{equation*}
x_k^i\to p^i,\quad \sup_{\varSigma}\tilde{u}_k\to-\infty \ for\ \, any\ \,\varSigma\subset M\setminus S
\end{equation*}
and
\begin{equation*}
2H_ke^{4\tilde{u}_k}\to \sum_{i=1}^m\beta_i\delta_{p^i} \ weakly\ \,in\ \,the\ \,sense\ \,of\ \,measures\ \,in\ \,M.
\end{equation*}
Futhermore, $\beta_i\in 16\pi^2\mathbb{N}$ if $p^i\notin\{q_1,\cdots,q_N\}$, and $\beta_i=16\pi^2(1+\gamma)$ if $p^i=q_j$ for some $j\in\{1,\cdots,N\}$.
\end{itemize}
\end{thm}
\begin{proof}[\textbf{Proof of Theorem \ref{thm-concentration-compactness}}]
We define $S$ to be the set of blow-up points of $\mathfrak{u}_k$ in $M$, that is,
\begin{equation*}
S=\{x\in M:\ \,\exists x_k\in M,\ \,{\rm s.t.}\ \,x_k\to x\ \, {\rm and}\ \,\tilde{u}_k(x_k)\to+\infty\ \,{\rm as}\ \,k\to+\infty\}.
\end{equation*}
We distinguish two cases.
\textbf{Case 1:} $S\neq\emptyset$.
For $p\in S$, Lemma \ref{lem-small-mass-regular} say that the mass of $\{\tilde{u}_k\}$ near $p$ is no less than $8\pi^2(1+\gamma)$.
Then finite integral assumption $\int_M H_ke^{4\tilde{u}_k}{\rm d}V_{g}\leq C$ implies ${\rm card}\,(S)\leq C$. We may denote $S=\{p_1,\cdots,p_m\}$ with some $n\in\mathbb{N}$. Therefore, there exists $r_0\in (0,1)$ such that for any $p^i\in S$, $p^i$ is the only blow-up point of $\tilde{u}_k$ in $B(p^i,r_0)$. Therefore, from the results in \cite{Druet-Robert} and Theorem \ref{thm-con-quan}, we obtain the alternative (iii).
\textbf{Case 2:} $S=\emptyset$.
In this case, we have $\sup_M\tilde{u}_k\leq C$, which implies $H_ke^{4\tilde{u}_k}$ is uniformly bounded in $M$. Taking into account of the Green's representation formula,
\begin{equation*}
\tilde{u}_k(x)-\bar{\tilde{u}}_k=\int_M G(x,y)H_k(y)e^{4\tilde{u}_k}{\rm d}V_{g}(y)=O(1).
\end{equation*}
Hence, after taking a subsequence, the alternative (i) occurs if $\limsup_{k\to+\infty}\int_M\tilde{u}_k{\rm d}V_{g}>-\infty$, the alternative (ii) holds if $\limsup_{k\to+\infty}\int_M\tilde{u}_k{\rm d}V_{g}\to-\infty$,
\end{proof}
Immediately, we derive the following two corollaries from Theorem \ref{thm-concentration-compactness}.
\begin{cor}
Suppose that $\tilde{u}_k$ satisfies the assumption in Theorem \ref{thm-concentration-compactness} and alternative (iii) occurs, then $\rho\in\Gamma$.
\end{cor}
\begin{cor}
Suppose that $\tilde{u}_k$ satisfies the assumption in Theorem \ref{thm-concentration-compactness}. Then for every $\rho\in\mathbb{R}^+\setminus\Gamma$, there exists a constant $C$ only depending on $\rho$, such that $\tilde{u}_k\leq C$. In particular, if we additionally assume $\int_M \tilde{u}_k{\rm d}V_g=0$ or $\int_M H_k e^{4\tilde{u}_k}{\rm d}V_g\geq c$ for some constant $c>0$, then
\begin{equation*}
\parallel\tilde{u}_k\parallel_{L^{\infty}(M)}\leq C.
\end{equation*}
\end{cor}
The following result explains that $\Gamma$ is some critical set to (\ref{Q-equ-gen-singular}) or (\ref{Q-equation-blowup}).
\begin{prop}[Critical Set]\label{critical-set}
Suppose {\rm Ker}\,$(P_g)=\{constants\}$ and that $\{u_k\}$ is a sequence of solutions to (\ref{Q-equation-blowup})$\sim$(\ref{volume-normal}) with the coefficients satisfying (\ref{assumption-coe}). If the blow-up phenomena occur, then $\int_M2b{\rm d}V_g\in\Gamma$.
\end{prop}
\begin{proof}[\textbf{Proof of Propertion \ref{critical-set} and Theorem \ref{thm-apriori-est}}]
Since $\int_MH_ke^{4\mathfrak{u}_k}{\rm d}V_{g}=\int_Mh_ke^{4u_k}{\rm d}V_g=\int_Mb_k{\rm d}V_g$ Proposition \ref{critical-set} and Theorem \ref{thm-apriori-est} obviously follow from the two corollaries above.
\end{proof}
\section{A spherical Harnack inequality}\label{harnack}
As a corollary of the Concentration-Compactness theorem, we derive a spherical Harnack inequality near a singular source $q_j$ when its strength $\gamma_i$ is not an integer. Recall that if $q_j$ is a blow-up point of $\mathfrak{u}_k$, then there exist $\delta>0$ and a sequence $\{x_k^j\}_{k\in\mathbb{N}}$ such that $\mathfrak{u}_k(x_k)=\max_{B(q_j,\delta)}\mathfrak{u}_k\to+\infty$ and $S\cap B(q_j,\delta)=\{q_j\}$. Thus, we have
\begin{equation*}
\varepsilon_k^j=\exp\Big(-\frac{\mathfrak{u}_k(x_k^j)}{1+\gamma_j}\Big)\to 0,\quad {\rm as}\;k\to+\infty.
\end{equation*}
\begin{cor}\label{lem-simple-blowup}
Suppose that $\tilde{u}_k$ satisfies the assumption in Theorem \ref{thm-concentration-compactness} and $q_j$ is a blow-up point of $\tilde{u}_k$. If $\gamma_j\notin \mathbb{N}$, then $d_{{g}}(x_k^j,q_j)=O(\varepsilon_k^j)$, that is
\begin{equation*}
y_k^j:=\frac{x_k^j-q_j}{\varepsilon_k^j}\to y^0\in\mathbb{R}^4
\end{equation*}
and $\tilde{u}_k$ has only one bubble at $q_j$.
\end{cor}
\begin{proof}[\textbf{Proof}]
Otherwise, we assume that
\begin{equation*}
\frac{x_k^j-q_j}{\varepsilon_k^j}\to+\infty\quad {\rm as }\ \,k\to+\infty.
\end{equation*}
For simplicity, we omit the superscript $j$ and work on the $\mathfrak{u}_k$ under the conformal normal coordinate. Let $r_k=d_{g}(x_k,q)$ and define the map $\varphi_k:B(0,\delta r_k^{-1})\to B(q,\delta)$ by
\begin{equation*}
\varphi_k:y\mapsto r_ky+q,
\end{equation*}
where on the right hand side we are using the conformal normal coordinates on $B(q,\delta)$. We use the notation $\breve{f}=\varphi_{k*}f=f\circ \varphi_k$ to denote the pull-back of a function $f$ defined on $B(q,\delta)$, and we let $\breve{g}_k=r_k^{-2}\varphi_{k*}g$ be the blow-up metric, i.e., a rescaling of the pull-back metric.
We define
\begin{equation*}
v_k(y)=\breve{\mathfrak{u}}_k+(1+\gamma)\log r_k,\quad y\in B(0,\delta/r_k).
\end{equation*}
Then $v_k$ satisfies
\begin{equation*}
\left\{\begin{array}{lcl}
P_{g_k}v_k+2\breve{\mathfrak{b}}_k=2\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k}, \quad {\rm in} \ \, B(0,\delta/r_k),\\
\int_{B(0,\delta/r_k)}\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k}{\rm d}V_{g_k}\leq C,
\end{array}
\right.
\end{equation*}
where $\breve{\mathfrak{b}}_k(y)=\mathfrak{b}_k(r_ky+q)$, $\breve{\mathfrak{h}}_k(y)=\mathfrak{h}_k(r_ky+q)$.
Denote $e_k=\frac{x_k-q}{r_k}$, then $e_k\to e_0\in S^1\subset\mathbb{R}^4$ and $e_0$ is a blow-up point of $\tilde{v}_k$. Applying Theorem \ref{thm-concentration-compactness}, there exists a finite blow-up set $\tilde{S}=\{y^1,\cdots,y^l\}$ such that $v_k\to-\infty$ uniformly on any compact subset of $\mathbb{R}^4\setminus \tilde{S}$ and
\begin{equation*}
2\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k}\to \sum_{i=1}^l\alpha_i\delta_{y^i},\quad {\rm weakly\ \,in\ \,the\ \,sense\ \,of\ \,measure},
\end{equation*}
where
\begin{equation*}
\alpha_i=\lim_{k\to+\infty}\int_{B(y^i,r_0)}2\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k(y)}{\rm d}V_{g_k}\quad{\rm and}\quad B(y^i,r_0)\cap \tilde{S}=\{y^i\}.
\end{equation*}
First, we claim that $0$ is not a blow-up point of $v_k$. Suppose $0\in\tilde{S}$, then by means of Theorem \ref{thm-concentration-compactness} we have
\begin{equation*}
\lim_{k\to+\infty}\int_{B(0,r_0)}\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k(y)}{\rm d}V_{g_k}=16\pi^2(1+\gamma).
\end{equation*}
On the other hand, since $e_0\in S^1$ is also a blow-up point of $v_k$ in $B(0,R)$ for $R$ large. Then
\begin{equation*}
\begin{split}
16\pi^2(1+\gamma)=&\lim_{k\to+\infty}\int_{B(q,\delta)}2\hat{h}_k(x)d_{g_k}(x,q)^{4\gamma}e^{4\mathfrak{u}_k(x)}{\rm d}V_{g_k} \\
=&\lim_{k\to+\infty}\int_{B(0,\delta/r_k)}2\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k(y)}{\rm d}V_{g_k} \\
\geq & \lim_{k\to+\infty}\int_{B(0,r_0)\cup B(e_0,r_0)}2\breve{\mathfrak{h}}_k(y)d_{g_k}(y,0)^{4\gamma}e^{4v_k(y)}{\rm d}V_{g_k} \\
=&16\pi^2(1+\gamma)+16\pi^2,
\end{split}
\end{equation*}
which yields a contradiction. Thus, $0\notin \tilde{S}$.
As a consequence, we have
\begin{equation*}
16\pi^2(1+\gamma)=16\pi^2l
\end{equation*}
which is impossible, since $\gamma\notin \mathbb{N}$ while $l$ is an integer. In other words, if $\gamma$ is not an integer, then $d_{g}(x_k,q)=O(\varepsilon_k)$ and $\hat{u}_k$ has only one bubble at $q$.
\end{proof}
\begin{thm}[A Spherical Harnack Inequality ($\gamma\notin\mathbb{N}$)]\label{SHT}
Suppose that $\tilde{u}_k$ satisfies the assumption in Theorem \ref{thm-concentration-compactness} and $q_j$ is a blow-up point of $\tilde{u}_k$. If $\gamma_j\notin \mathbb{N}$, then near $q_j$ there holds the following spherical Harnack inequality:
\begin{equation}\label{spherical-Harnack}
\max_{x\in B(q_j,\delta)}\{\tilde{u}_k(x)+(1+\gamma_j)\log|x-q|\}\leq C
\end{equation}
with some constant $C$.
\end{thm}
\begin{proof}[\textbf{Proof}]
Suppose that (\ref{spherical-Harnack}) fails, then there exists a sequence $\{x_k\}\subset B(q_j,\delta)$ such that $\max_{x\in B(q_j,\delta)}\{\tilde{u}_k(x)+(1+\gamma_j)\log|x-q|\}\to +\infty$. By means of the selection process and quantization property in \cite{Malchiodi,Druet-Robert}, we obtain that $\int_{B(q,\delta)}H_k(x)e^{4\tilde{u}_k(x)}{\rm d}V_{g}\to16\pi^2n$ for some positive integer $n$. However, from Theorem \ref{thm-concentration-compactness}, we know that
\begin{equation}
\int_{B(q,\delta)}H_k(x)e^{4\tilde{u}_k(x)}{\rm d}V_{g}\to16\pi^2(1+\gamma),\quad as\ \, k\to+\infty,
\end{equation}
which is impossible.
\end{proof}
\section*{Appendix: Comparsion between $d_g(x,q)$ and $|x-q|$}
\setcounter{equation}{0}
\setcounter{subsection}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
\renewcommand{\thesubsection}{A.\arabic{subsection}}
In this appendix, we will establish the comparison between the distance $d_g(x,q)$ and its derivatives and their Euclidean counterparts as in the Appendix B in \cite{zhang-weinstein}. We will follow the argument in \cite{zhang-weinstein} and give the detail for completeness. We claim that for $j=0,1,2,3$, there holds
\begin{equation}\label{comparsion-dist}
\nabla^j\big(\log|x-y|-\log d_g(x,y)\big)=O(r^{2-j}),\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation}
We recall that $g$ is the conformal normal metric centered at $q$, and we identity $x,y\in T_qM$ with $\exp_qx$ and $\exp_qy$ respectively, where $\exp_q$ is the exponential map at $q$ with respect to the metric $g$. Thus,
\begin{equation*}
d_g(x,y)=d(\exp_qx,\exp_qy), \quad x,y\in B(q,\delta).
\end{equation*}
Also, we denote $\nabla_{g}$ by $\nabla$ for convenience. First, let us note that the following simple estimates on $d$ hold:
\begin{equation}\label{rough-comparsion-dist}
\big|\nabla^j(\log d_g(x,y))\big|\leq C|x-y|^{-j},\quad j=1,2,3,4.
\end{equation}
Set
\begin{equation*}
f(x)=\log|x-y|-\log d_g(x,y),\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation*}
We aim to show that
\begin{equation*}
\big|\nabla^jf(x)\big|\leq C|x-q|^{2-j},\quad j=0,1,2,3,\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation*}
Let $R$, $R_{ij}$ and $R_{ijkl}$ respectively denote the scalar, Ricci and Riemann curvature of $g$. From the definitions of $g$ and $R^i_{jkl}$, we obtain that
\begin{equation*}
\nabla^jR^i_{jkl}(x)=O(1),\quad j=1,2.
\end{equation*}
In conformal normal coordinates, there holds $R(q)=R_{ij}(q)=|\nabla R(q)|=0$. As a consequence, we have futher
\begin{equation*}
R(x)=O(r^2),\quad R_{ij}(x)=O(r).
\end{equation*}
We shall derive an estimate on $\Delta_{g}^2f(x)$. By the definition of $g$ and (A.4) in \cite{zhang-weinstein}, that is
\begin{equation*}
P_{g,y}\Big(-\frac{1}{8\pi^2}\chi(r)\log d_g(x,y)\Big)=\delta_x+E(x,y),\quad {\rm with \ \,}E \ \,{\rm bounded},
\end{equation*}
we have that
\begin{equation}
P_{g}\log d_g(x,q)=O(r^4),\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation}
In view of the rough estimates (\ref{rough-comparsion-dist}), we can estimate the term:
\begin{equation*}
\big(P_{g}-\Delta_{g}^2\big)\log d_g(x,q)=\partial_m\Big(g^{mi}\big(\frac{2}{3}R(x)g_{ij}-2R_{ij}(x)\big)g^{lj}\partial_j\big(\log d_g(x,q)\big)\Big)=O(r^{-1}).
\end{equation*}
Therefore, we can get
\begin{equation*}
\Delta_{g}^2\big(\log d_g(x,q)\big)=O(r^{-1}),\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation*}
Finally, we consider the term $\Delta_{g}^2\big(\log |x-q|\big)$. Since $\Delta^2\big(\log |x-q|\big)=0$, it suffices to estimate $\Delta_{g}^2-\Delta^2$. For any function $u$, the direct computation leads to
\begin{equation}\label{expansion-4-order}
\begin{split}
\Delta_{g}^2u=&g^{ab}g^{ij}\partial_{ijab}u+2\partial_{ija}u\big(\partial_bg^{ab}g^{ij}+g^{ab}\partial_bg^{ij}\big) \\
&+\partial_{ij}u\big(\partial_ag^{ab}\partial_bg^{ij}+2g^{ai}\partial_{ab}g^{bj}+g^{ab}\partial_{ab}g^{ij}+\partial_ag^{ia}\partial_bg^{bj}\big) \\
&+\partial_ju\big(\partial_ag^{ab}\partial_{ib}g^{ij}+g^{ab}\partial_{iab}g^{ij}\big),
\end{split}
\end{equation}
where we have used $\det\,(g)=1$. Using the expansion of $g^{ab}$:
\begin{equation*}
g^{ab}(x)=\delta_{ab}-\frac{1}{3}R_{mabl}(q)x^ax^b+O(|x-q|^3),
\end{equation*}
and replacing $u$ by $\log|x-q|$ in (\ref{expansion-4-order}) above, we obtain that
\begin{equation*}
\Delta_{g}^2\big(\log|x-q|\big)=O(r^{-2}).
\end{equation*}
Consequently,
\begin{equation}\label{app-equ}
\Delta_{g}^2 f(x)=O(r^{-2}),\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation}
An estimate on the $L^{\infty}$-norm of $f(x)$ can easily be seen as follows:
\begin{equation}
d_g(x,q)=\int_{0}^{x(t)}\sqrt{x_i'(t)g_{ij}(t)x_j'(t)}{\rm d}t=|x-q|\big(1+O(r^2)\big),
\end{equation}
which implies
\begin{equation}\label{app-C0}
f(x)=O(r^2),\quad x\in B(q,2r)\setminus B(q,r/2).
\end{equation}
Applying the elliptic theory to (\ref{app-equ}) and (\ref{app-C0}), we obtain the claim (\ref{comparsion-dist})
and
\begin{equation}\label{0-comparsion}
d_g(x,q)=|x-q|(1+O(r^2)).
\end{equation}
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BkiUd0E5qhDBdMC-Fcd6 | \section{Supplemental Material: \\ Sign-changing photon-mediated atomic interactions in multimode cavity QED}
\subsection{Spectrum of a confocal cavity}
Within paraxial optics, the beam inside a Fabry-Perot cavity is described by Hermite-Gaussian modes. A mode $\Phi_{Q,l,m}$ is labeled by one longitudinal index $Q$ and two transverse indices $l$ and $m$. These indices count the number of field nodes along their respective axes. For a symmetric two-mirror cavity of length $L$, with $R$ as the mirror radius of curvature, the frequency of a given mode is
\begin{equation}
f_{Qlm}=\frac{c}{2L}\big[ Q + \frac{l+m+1}{\pi} \arccos{ g} \big],
\end{equation}
where $c$ is the speed of light inside the cavity, $g = 1-L/R$ and $c/2L$ is the free spectral range of the cavity. The term proportional to $\arccos{g}$ captures the effect of additional Gouy phase shifts on higher-order transverse modes,
which involve terms proportional to $(l+m+1)\psi(z)$, where $\psi(z) = \mathrm{arctan} (z/z_R)$ is the Gouy phase and $z_R$ is the Rayleigh range $z_R = \pi w_0^2/\lambda$.
In general, different transverse modes will be resonant at different frequencies; however, degenerate cavities with special geometries can support a family of transverse modes, each with distinct spatial profiles, at a single frequency. In particular, a confocal cavity has $L=R$ and thus $g = 0$. Therefore, all modes that satisfy the condition
\begin{equation}
Q + \frac{1}{2}(l + m + 1) = Q_0 + \frac{(\eta+1)}{2}
\label{rescondition}
\end{equation}
will be resonant at the same frequency $c(2 Q_0+\eta+1)/4L$, where $Q_0$ is a positive integer and $\eta=0 (1)$ for even (odd) families. At every half free spectral range, the transverse mode content varies between all even modes $l+m~\mathrm{mod}~2 = 0$ and all odd modes $l+m~\mathrm{mod}~2 = 1$. Within a degenerate resonance, to satisfy Eq.~\eqref{rescondition}, different transverse modes must carry different longitudinal indices. This causes the longitudinal profile of sequential transverse modes within a degenerate resonance to cycle between $+\cos{k_r z}$, $-\sin{k_r z}$, $-\cos{k_r z}$, and $+\sin{k_r z}$, as described in Fig.~\ref{fig1}(a) of the main text.
\subsection{Experimental apparatus}
This work employs a $R=1$-cm radius-of-curvature confocal cavity of length $L=R$. The length of the multimode cavity is adjustable~\cite{Kollar2015}, though in this work we set $L=R$. We trap within this cavity a BEC of $2.5(3) {\times} 10^5$ $\mathrm{Rb}^{87}$ atoms in the $|F=1,m_F=-1 \rangle$ state. See Ref.~\cite{Kollar2015} for BEC preparation procedure and Fig.~\ref{fig1} for illustration of experiment. The BEC is confined in a crossed optical dipole trap (ODT) formed by a pair of $1064$-nm laser beams propagating along $\hat{x}$ and $\hat{y}$ with waists of $40$~$\mu$m in the $xy$-plane and $80$~$\mu$m along $\hat{z}$. The resulting trap frequencies of $(\omega_x,\omega_y,\omega_z) = 2 \pi \times [189(2),134(1),90(1)]$~Hz create a compact BEC with Thomas-Fermi radii $(R_x, R_y, R_z) = [4.2(1), 5.8(3), 8.9(1)]$~$\mu$m that are significantly smaller than the $w_0 = 35$~$\mu$m waist of the TEM$_{0,0}$ cavity mode. Acousto-optic deflectors (AODs) placed in the path of each ODT control the intensity and location of the ODTs, allowing us to translate the BEC to any point in the $xy$-plane with an uncertainty of $0.9~\mu$m. In the experiments of Figs.~\ref{fig3} and~\ref{fig4}, we use dynamic trap shaping~\cite{Henderson09} to produce two smaller BECs of $2.0(3) {\times} 10^5$ atoms each, with a population imbalance uncertainty of ${<}10$\%. The relative position of these BECs along $\hat{x}$ is controlled by the AOD.
Both the local oscillator beam (used for holographic imaging of the cavity emission) and the transverse pump are derived from the same laser but pass through different acousto-optic modulators (AOMs) for intensity stabilization. To maintain the relative phase stability between the two beams, both AOMs are driven by signals from the same multichannel direct digital synthesizer. This synthesizer is synced to a stable Rb frequency reference. Due to path length drift, the relative phase between the pump and the local oscillator is stable only within the same experiment realization.
\subsection{Holographic imaging}
The employed holographic imaging method is described in detail in Ref.~\cite{Kroeze:2018wd} and is similar to that reported in Ref.~\cite{Schine:2018ui}. Briefly, a portion of the pump field---serving as a local oscillator (LO)---is directed onto the same EMCCD camera onto which the cavity emission is imaged. The cavity field $E_c(\mathbf{r}) = |E_c(\mathbf{r})|e^{i \phi_c(\mathbf{r})}$ and the LO field $E_\text{LO}(\mathbf{r}) = |E_\text{LO}(\mathbf{r})|e^{i \phi_\text{LO}(\mathbf{r})}$ interfere to form a spatial heterodyne image $I_h(\mathbf{r})$. The image's interference fringes are proportional to the phase and amplitude of the cavity field:
\begin{equation}
I_{h}(\mathbf{r}) \propto |E_c(\mathbf{r})E_\text{LO}(\mathbf{r})| \cos \left[ \Delta \mathbf{k} \cdot \mathbf{r} + \Delta\phi(\mathbf{r}) \right],
\label{hologram}
\end{equation}
where the phase difference between the cavity and LO wavefronts is $\Delta\phi(\mathbf{r}) =\phi_c(\mathbf{r}) -\phi_\text{LO}(\mathbf{r}) $. The amplitude and phase of the fringes produced are a measure of $|E_c(\mathbf{r})|$ and $\phi_c(\mathbf{r})$.
Demodulating this image at the fringe wavevector $\Delta \mathbf{k}$ provides a holographic reconstruction of $|E_c(\mathbf{r})|$ and $\phi_c(\mathbf{r})$. Accurate extraction of these images requires the correction of LO intensity and phase variation. To do so for the confocal cavity, we perform a least-squares fit to the cavity emission intensity pattern using the exact theory result from Ref.~\cite{Vaidya:2018fp}. We extract the LO phase variation from the difference between measured phase and the expected phase.
\subsection{Effective Hamiltonian}
The Green's function for the cavity-mediated interaction in a perfect confocal cavity near an even degenerate resonance can be written as a sum of the contributions from the two classes of longitudinal modes~\cite{Vaidya:2018fp,GouyPRA2018}:
\begin{align}
4\mathcal{D}^{+}(\mathbf{x},\mathbf{x}') = 4\mathcal{D}^{+} (\mathbf{r},\mathbf{r^\prime},z,z^\prime) &=D_c (\mathbf{r},\mathbf{r^\prime})\cos{k_r z} \cos{k_r z^\prime} \nonumber \\
&+ D_{s} (\mathbf{r},\mathbf{r^\prime}) \sin{k_r z} \sin{k_r z^\prime},
\end{align}
with
\begin{equation}
\begin{cases}
D_{c} = \delta\Big(\frac{\sqrt{2}(\mathbf{r} - \mathbf{r^\prime})}{w_0}\Big) + \delta\Big(\frac{\sqrt{2}(\mathbf{r} + \mathbf{r^\prime})}{w_0}\Big) + \frac{1}{\pi} \cos\big(\frac{2 \mathbf{r} \cdot \mathbf{r^\prime}}{w^2_0}\big) \\
D_{s} = \delta\Big(\frac{\sqrt{2}(\mathbf{r} - \mathbf{r^\prime})}{w_0}\Big) + \delta\Big(\frac{\sqrt{2}(\mathbf{r} + \mathbf{r^\prime})}{w_0}\Big) - \frac{1}{\pi} \cos\big(\frac{2 \mathbf{r} \cdot \mathbf{r^\prime}}{w^2_0}\big).
\end{cases}
\end{equation}
To allow for the full phase freedom in the atomic density wave, the atomic profile is expanded as
\begin{eqnarray}
\Psi(\mathbf{x}) &&=\sqrt{\rho(\mathbf{r})}\times \\\nonumber
&&\big[ \psi_0 +\sqrt{2}\cos{k_rx}( \psi_c \cos{k_rz} + \psi_s \sin{k_rz}) \big],
\end{eqnarray}
where for simplicity we shall assume a $\delta$-function transverse atomic profile $\rho(\mathbf{r}) = \delta(\mathbf{r} - \mathbf{r}_0)$ since the Thomas-Fermi radius of the BEC is much smaller than the cavity waist $w_0$, $\mathbf{r}_0$ is the location of the atoms in the cavity transverse plane, $\psi_0$ is the ground state fraction of the gas that has a uniform density profile (compared to the $\lambda$-scale) and $\psi_{c (s)}$ is the excited atomic density wave in the $\cos{k_rz}~(\sin{k_rz})$ pattern. The Hamiltonian is then
\begin{align}\label{Ham}
H= E_0 \int &d^3\mathbf{x} d^3\mathbf{x^\prime} \cos(k_r x) \cos(k_r x') \times \nonumber \\
&|\Psi(\mathbf{x})|^2 \mathcal{D}^{+}(\mathbf{x},\mathbf{x^\prime}) |\Psi(\mathbf{x^\prime})|^2 \equiv -E_0 \mathcal{H},
\end{align}
where $E_0$ is a positive constant prefactor, and $\cos(k_r x) \cos(k_r x')$ term is due to the standing wave pump. Focusing only on the terms involving $\cos(2 \mathbf{r} \cdot \mathbf{r^\prime}/w^2_0)$ in $\mathcal{D}^{+}(\mathbf{x},\mathbf{x}')$, the effective Hamiltonian $\mathcal{H}$ can then be evaluated as
\begin{equation}
\mathcal{H} = -\frac{1}{8 \pi}\left[ |\psi_0 \psi^{*}_{c} + \psi^{*}_0 \psi_{c}|^2 - |\psi_0 \psi^{*}_{s} + \psi^{*}_0 \psi_{s}|^2 \right] \cos\left(\frac{2 r^2_0}{w_0}\right).
\end{equation}
Defining the following order parameters
\begin{align}
\chi_c &= \frac{\psi_0 \psi^{*}_{c} + \psi^{*}_0 \psi_{c}}{N} \nonumber \\
\chi_s &= \frac{\psi_0 \psi^{*}_{s} + \psi^{*}_0 \psi_{s}}{N},
\end{align}
and ignoring the numeric prefactor, we recover the effective Hamiltonian in the main text, where $N$ is the total atom number. For two BECs, the cross term in the integral in Eq.~\ref{Ham} gives rise to the interaction term
\begin{equation}
H_{12} \propto -J_{12} (\chi_{c1} \chi_{c2} - \chi_{s1} \chi_{s2}),
\end{equation}
where $J_{12} = 2N\cos\left({2\mathbf{r_1} \cdot \mathbf{r_2}}/{w^2_0}\right)$.
\end{document}
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End of preview. Expand
in Data Studio
Annotated SlimPajama Dataset
Dataset Description
This dataset contains the first fully annotated SlimPajama dataset with comprehensive quality metrics for data-centric large language model research. The dataset includes approximately 580 billion tokens from the training set of the original SlimPajama dataset, annotated across 25 different quality dimensions.
Note: This dataset contains only the training set portion of the original SlimPajama dataset, which is why the token count is approximately 580B rather than the full 627B tokens.
Dataset Statistics
- Total samples: ~580B tokens from SlimPajama training set
- Quality metrics: 25 dimensions across 3 categories
- Domains: 7 domains (CommonCrawl, C4, GitHub, Books, ArXiv, Wikipedia, StackExchange)
- Annotation coverage: 100% of the training set
Quality Metrics
The dataset includes 25 quality scores across three main categories:
1. Natural Language Quality Signals (11 metrics)
Rule-based measures from RedPajama indicating text naturalness:
rps_doc_frac_no_alph_words: Fraction of words with no alphabetical charactersrps_doc_mean_word_length: Mean word length after normalizationrps_doc_frac_unique_words: Fraction of unique words (degeneracy measure)rps_doc_unigram_entropy: Entropy of unigram distributionrps_doc_word_count: Number of words after normalizationrps_lines_ending_with_terminal_punctution_mark: Lines ending with terminal punctuationrps_lines_numerical_chars_fraction: Ratio of numerical to total charactersrps_lines_uppercase_letter_fraction: Ratio of uppercase to total charactersrps_doc_num_sentences: Number of sentences in contentrps_doc_frac_chars_top_2gram: Fraction of characters in top word 2-gramrps_doc_frac_chars_top_3gram: Fraction of characters in top word 3-gram
2. Data Importance Scores (3 metrics)
DSIR-based importance weights measuring similarity to high-quality domains:
dsir_books: Importance score relative to Books domaindsir_wiki: Importance score relative to Wikipedia domaindsir_math: Importance score relative to AutoMathText domain
3. Model-based Quality Ratings (11 metrics)
Existing Metrics:
fineweb_edu: Educational value (from FineWeb-Edu) - single value in list formatad_en: Advertisement detection (from WanjuanCC) - logits for binary classification [label_0, label_1]fluency_en: Fluency assessment (from WanjuanCC) - logits for binary classification [label_0, label_1]qurater: QuRating scores as a list [Writing Style, Required Expertise, Facts and Trivia, Educational Value]
PRRC Framework (Our Contribution):
modernbert_professionalism: Professionalism logits for 6 levels (0-5 scale) - use argmax() to get ratingmodernbert_readability: Readability logits for 6 levels (0-5 scale) - use argmax() to get ratingmodernbert_reasoning: Reasoning logits for 6 levels (0-5 scale) - use argmax() to get ratingmodernbert_cleanliness: Cleanliness logits for 6 levels (0-5 scale) - use argmax() to get rating
PRRC Framework Details
Our PRRC framework introduces four novel dimensions for comprehensive data quality assessment:
- Professionalism: Measures the degree of expertise and prerequisite knowledge required to comprehend the text
- Readability: Evaluates text clarity, coherence, and ease of understanding
- Reasoning: Assesses the complexity of logical reasoning and analytical thinking required
- Cleanliness: Evaluates text formatting, completeness, and absence of noise/irrelevant content
Each PRRC dimension uses a 5-point additive rating system, with models achieving F1 scores of 87-92% on test sets.
Dataset Structure
The dataset structure for each example:
{
"id": "unique_document_id",
"content": "Main text content of the document",
"sub_path": "domain_name", # e.g., "arxiv", "github", "wikipedia", etc.
# Natural Language Quality Signals (RedPajama-style metrics)
"rps_doc_frac_no_alph_words": float,
"rps_doc_mean_word_length": float,
"rps_doc_frac_unique_words": float,
"rps_doc_unigram_entropy": float,
"rps_doc_word_count": int,
"rps_lines_ending_with_terminal_punctution_mark": float,
"rps_lines_numerical_chars_fraction": float,
"rps_lines_uppercase_letter_fraction": float,
"rps_doc_num_sentences": int,
"rps_doc_frac_chars_top_2gram": float,
"rps_doc_frac_chars_top_3gram": float,
# Data Importance Scores (DSIR)
"dsir_books": float,
"dsir_wiki": float,
"dsir_math": float,
# Model-based Quality Ratings
"fineweb_edu": [float], # Single value in list
"ad_en": [float, float], # [has_ad_logit, no_ad_logit] - use argmax() to get 0-1 rating
"fluency_en": [float, float], # [not_fluent_logit, fluent_logit] - use argmax() to get 0-1 rating
"qurater": [float, float, float, float], # [Writing Style, Required Expertise, Facts and Trivia, Educational Value]
# PRRC Framework (Our Contribution) - all contain 6 logits for levels 0-5
"modernbert_professionalism": [float, float, float, float, float, float], # Use argmax() to get 0-5 rating
"modernbert_readability": [float, float, float, float, float, float], # Use argmax() to get 0-5 rating
"modernbert_reasoning": [float, float, float, float, float, float], # Use argmax() to get 0-5 rating
"modernbert_cleanliness": [float, float, float, float, float, float] # Use argmax() to get 0-5 rating
}
Usage
Loading the Dataset
from datasets import load_dataset
# Load the full dataset
dataset = load_dataset("opendatalab/SlimPajama-627B-Annotated")
# Load a specific split if available
train_dataset = load_dataset("opendatalab/SlimPajama-627B-Annotated", split="train")
Data Processing and Selection Example
import pandas as pd
import numpy as np
from datasets import load_dataset
# Load dataset
dataset = load_dataset("opendatalab/SlimPajama-627B-Annotated", split="train")
# Convert to pandas for easier manipulation
df = dataset.to_pandas()
# Process PRRC scores (convert logits to ratings using argmax)
df['professionalism_score'] = df['modernbert_professionalism'].apply(lambda x: np.argmax(x))
df['readability_score'] = df['modernbert_readability'].apply(lambda x: np.argmax(x))
df['reasoning_score'] = df['modernbert_reasoning'].apply(lambda x: np.argmax(x))
df['cleanliness_score'] = df['modernbert_cleanliness'].apply(lambda x: np.argmax(x))
# Process binary classification scores
df['advertisement_score'] = df['ad_en'].apply(lambda x: np.argmax(x)) # 0 = has ad, 1 = no ad
df['fluency_score'] = df['fluency_en'].apply(lambda x: np.argmax(x)) # 0 = not fluent, 1 = fluent
# Extract QuRating scores
df['writing_style'] = df['qurater'].apply(lambda x: x[0])
df['required_expertise'] = df['qurater'].apply(lambda x: x[1])
df['facts_trivia'] = df['qurater'].apply(lambda x: x[2])
df['educational_value'] = df['qurater'].apply(lambda x: x[3])
# Extract FineWeb-Edu score
df['fineweb_educational'] = df['fineweb_edu'].apply(lambda x: x[0])
# Example: Multi-dimensional quality score combination (Meta-rater approach)
# Using the learned weights from the Meta-rater paper
weights = {
'educational_value': 0.0564, # From qurater[3]
'rps_doc_frac_no_alph_words': 0.0493,
'fineweb_educational': 0.0493,
'rps_lines_uppercase_letter_fraction': 0.0488,
'facts_trivia': 0.0477, # From qurater[2]
'rps_doc_frac_chars_top_3gram': 0.0473,
'rps_lines_ending_with_terminal_punctution_mark': 0.0473,
'rps_doc_frac_chars_top_2gram': 0.0471,
'dsir_wiki': 0.0469,
'rps_lines_numerical_chars_fraction': 0.0460,
'rps_doc_num_sentences': 0.0458,
'dsir_math': 0.0448,
'reasoning_score': 0.0444,
'rps_doc_frac_unique_words': 0.0432,
'rps_doc_word_count': 0.0423,
'rps_doc_unigram_entropy': 0.0422,
'dsir_books': 0.0414,
'professionalism_score': 0.0405,
'fluency_score': 0.0402,
'readability_score': 0.0393,
'required_expertise': 0.0373, # From qurater[1]
'advertisement_score': 0.0368,
'cleanliness_score': 0.0117,
'rps_doc_mean_word_length': 0.0065,
'writing_style': 0.0005, # From qurater[0]
}
# Calculate weighted quality score
quality_score = np.zeros(len(df))
for metric, weight in weights.items():
if metric in df.columns:
quality_score += df[metric].values * weight
# Select top-k samples based on quality score
top_k = 10000
top_k_indices = np.argsort(quality_score)[-top_k:]
selected_data = df.iloc[top_k_indices]
print(f"Selected top {top_k} samples using Meta-rater weights")
Applications
This annotated dataset enables:
- Data-Centric LLM Research: Study the impact of different quality dimensions on model performance
- Multi-dimensional Data Selection: Implement sophisticated data selection strategies beyond single-metric approaches
- Quality Score Analysis: Analyze correlations and relationships between different quality metrics
- Benchmark Development: Create standardized benchmarks for data quality assessment
- Efficient Pre-training: Select high-quality subsets for more efficient model training
- Domain-specific Analysis: Compare quality distributions across different domains (ArXiv, GitHub, Wikipedia, etc.)
Annotation Process
The quality scores were generated using:
- Rule-based metrics: Extracted using established heuristics from RedPajama and DSIR
- Existing model-based ratings: Applied pre-trained classifiers from FineWeb-Edu, WanjuanCC, and QuRating
- PRRC ratings: Generated using Llama-3.3-70B-Instruct for annotation, followed by fine-tuned ModernBERT models for efficient scoring
📚 Citation
If you use Meta-rater in your research, please cite our paper:
@article{zhuang2025meta,
title={Meta-rater: A Multi-dimensional Data Selection Method for Pre-training Language Models},
author={Zhuang, Xinlin and Peng, Jiahui and Ma, Ren and Wang, Yinfan and Bai, Tianyi and Wei, Xingjian and Qiu, Jiantao and Zhang, Chi and Qian, Ying and He, Conghui},
journal={arXiv preprint arXiv:2504.14194},
year={2025}
}
📄 License
This dataset is released under the same license as the original SlimPajama dataset. Please refer to the original SlimPajama repository for licensing details.
🤝 Acknowledgments
This work builds upon:
- SlimPajama: The original dataset from Cerebras
- RedPajama: Natural language quality signals
- DSIR: Data importance scoring methodology
- FineWeb-Edu: Educational value assessment
- WanjuanCC: Advertisement and fluency detection
- QuRating: Multi-dimensional quality rating framework
📞 Contact
- Project Lead: Ren Ma ([email protected])
- Corresponding Author: Conghui He ([email protected])
- Issues: Please use GitHub Issues for questions.
⭐ Star us on GitHub and HuggingFace if you find Meta-rater useful! ⭐
Made with ❤️ by the OpenDataLab team
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